Some Standard Continuous Distributions

Gauranga C Samanta

Associate Professor, P. G. Department of Mathematics

Fakir Mohan University, Balasore-756019, Odisha

August 16, 2022

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 1 / 45
 Outline

(i) Uniform Distribution

(ii) Gamma Distribution
(iii) Exponential Distribution
(iv) Chi-Square Distribution
(v) Normal Distribution

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 Uniform Distribution

Definition 1.
A random variable X is said to be uniform on the interval [a, b] if its
1
probability density function is of the form f (x) = b−a , a ≤ x ≤ b, where
a and b are constants
Note: We denote a random variable X with the uniform distribution on
the interval [a, b] as X ∼ U(a, b).
0, if x ≤ a

The cdf of X is given by F (x) = x−a b−a , if a < x < b

1, if , x ≥ b


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 Uniform Distribution Cont.

Theorem 2.
Let X be uniformly destributed over an interval (a, b) then
a+b
1. E [X ] = 2
2
2. Var (X ) = (b−a)
12
( tb ta
e −e
t(b−a) , if t 6= 0
3. MX (t) =
1, if t = 0

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 Uniform Distribution Cont.

Example 3.
Suppose Y ∼ U(0, 1) and Y = X 2 . What is the probability density
function of X ?

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 Uniform Distribution Cont.

Example 3.
Suppose Y ∼ U(0, 1) and Y = X 2 . What is the probability density
function of X ?
(
x
,0 ≤ x ≤ 2
ANS: f (x) = 2
0, otherwise

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 5 / 45
 Uniform Distribution Cont.

Example 3.
Suppose Y ∼ U(0, 1) and Y = X 2 . What is the probability density
function of X ?
(
x
,0 ≤ x ≤ 2
ANS: f (x) = 2
0, otherwise

Example 4.
10
If X ∼ U(0, 10), then what is P(X + X ≥ 7)?

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 5 / 45
 Uniform Distribution Cont.

Example 3.
Suppose Y ∼ U(0, 1) and Y = X 2 . What is the probability density
function of X ?
(
x
,0 ≤ x ≤ 2
ANS: f (x) = 2
0, otherwise

Example 4.
10
If X ∼ U(0, 10), then what is P(X + X ≥ 7)?
7
ANS: 10

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 Uniform Distribution Cont.

Example 5.
If X ∼ U(0, 3), what is the probability that the quadratic equation
4t 2 + 4tX + X + 2 = 0 has real solutions?

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 Uniform Distribution Cont.

Example 5.
If X ∼ U(0, 3), what is the probability that the quadratic equation
4t 2 + 4tX + X + 2 = 0 has real solutions?

ANS 0.333

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 Uniform Distribution Cont.

Example 5.
If X ∼ U(0, 3), what is the probability that the quadratic equation
4t 2 + 4tX + X + 2 = 0 has real solutions?

ANS 0.333
Theorem 6.
If X is a continuous random variable with a strictly increasing cumulative
distribution function F (x), then the random variable Y , defined by
Y = F (X ) has the uniform distribution on the interval [0, 1].

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 6 / 45
 Uniform Distribution Cont.

Example 7.
If the probability density function of X is
e −x 1
f (x) = (1+e −x )2 , − ∞ < x < ∞, then what is the pdf of Y = 1+e −X
?

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 Uniform Distribution Cont.

Example 7.
If the probability density function of X is
e −x 1
f (x) = (1+e −x )2 , − ∞ < x < ∞, then what is the pdf of Y = 1+e −X
?

ANS: Y ∼ U(0, 1)

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 7 / 45
 Uniform Distribution Cont.

Example 7.
If the probability density function of X is
e −x 1
f (x) = (1+e −x )2 , − ∞ < x < ∞, then what is the pdf of Y = 1+e −X
?

ANS: Y ∼ U(0, 1)

Example 8.
A box to be constructed so that its height is 10 inches and its base is X
inches by X inches. If X has a uniform distribution over the interval (2, 8),
then what is the expected volume of the box in cubic inches?

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 Uniform Distribution Cont.

Example 7.
If the probability density function of X is
e −x 1
f (x) = (1+e −x )2 , − ∞ < x < ∞, then what is the pdf of Y = 1+e −X
?

ANS: Y ∼ U(0, 1)

Example 8.
A box to be constructed so that its height is 10 inches and its base is X
inches by X inches. If X has a uniform distribution over the interval (2, 8),
then what is the expected volume of the box in cubic inches?

ANS: 280 cubic inches

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 Uniform Distribution Cont.

Example 9.
Two numbers are chosen independently and at random from the interval
(0, 1). What is the probability that the two numbers differs by more than
1
2?

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 Uniform Distribution Cont.

Example 9.
Two numbers are chosen independently and at random from the interval
(0, 1). What is the probability that the two numbers differs by more than
1
2?

1
ANS: 4

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 Uniform Distribution Cont.

Example 9.
Two numbers are chosen independently and at random from the interval
(0, 1). What is the probability that the two numbers differs by more than
1
2?

1
ANS: 4

Example 10.
If X ∼ U(0, 1), then find E [e X ]

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 Uniform Distribution Cont.

Example 9.
Two numbers are chosen independently and at random from the interval
(0, 1). What is the probability that the two numbers differs by more than
1
2?

1
ANS: 4

Example 10.
If X ∼ U(0, 1), then find E [e X ]

ANS e − 1

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 Uniform Distribution Conti.
Example 11.
If a stick of length 1 unit is split at a point u that is uniformly distributed
over (0, 1), determine the expected length of the piece that contains the
point 0 ≤ x ≤ 1.

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 Uniform Distribution Conti.
Example 11.
If a stick of length 1 unit is split at a point u that is uniformly distributed
over (0, 1), determine the expected length of the piece that contains the
point 0 ≤ x ≤ 1.
1
ANS 2 + x(1 − x)

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 Uniform Distribution Conti.
Example 11.
If a stick of length 1 unit is split at a point u that is uniformly distributed
over (0, 1), determine the expected length of the piece that contains the
point 0 ≤ x ≤ 1.
1
ANS 2 + x(1 − x)
Example 12.
If a point x is taken at random on a line AB of length 2a, with all
positions of the point being equally likely, find the expected value of the
rectangle AxxB.

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 Uniform Distribution Conti.
Example 11.
If a stick of length 1 unit is split at a point u that is uniformly distributed
over (0, 1), determine the expected length of the piece that contains the
point 0 ≤ x ≤ 1.
1
ANS 2 + x(1 − x)
Example 12.
If a point x is taken at random on a line AB of length 2a, with all
positions of the point being equally likely, find the expected value of the
rectangle AxxB.

ANS 23 a2

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 Uniform Distribution Conti.
Example 11.
If a stick of length 1 unit is split at a point u that is uniformly distributed
over (0, 1), determine the expected length of the piece that contains the
point 0 ≤ x ≤ 1.
1
ANS 2 + x(1 − x)
Example 12.
If a point x is taken at random on a line AB of length 2a, with all
positions of the point being equally likely, find the expected value of the
rectangle AxxB.

ANS 23 a2
Example 13.
If a strig of length 1 meter is cut into 2 pieces at a random point along its
length, what is the probability that the longer piece is at least twice the
length of the shorter one?
Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 9 / 45
 Uniform Distribution Conti.
Example 11.
If a stick of length 1 unit is split at a point u that is uniformly distributed
over (0, 1), determine the expected length of the piece that contains the
point 0 ≤ x ≤ 1.
1
ANS 2 + x(1 − x)
Example 12.
If a point x is taken at random on a line AB of length 2a, with all
positions of the point being equally likely, find the expected value of the
rectangle AxxB.

ANS 23 a2
Example 13.
If a strig of length 1 meter is cut into 2 pieces at a random point along its
length, what is the probability that the longer piece is at least twice the
length of the shorter one?
Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 9 / 45
 Gamma Distribution

Note: The gamma distribution involves the notion of gamma function.

First, we develop the notion of gamma function and study some of its well
known properties.

Definition 14 (Gamma Function).

The gamma function is defined as
Z ∞
Γ(z) = x z−1 e −x dx,
0

where z is positive real number. The condition z > 0 is assumed for the
convergence of the integral.

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 Gamma Distribution Cont.

Some Results on Gamma Function:

1. If n is a natural number, then Γ(n + 1) = n!
2. Γ(1) = 1

√
3. Γ 12 = π
√
4. Γ − 21 = −2 π

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 Gamma Distribution Cont.

Definition 15.
1 −x
α−1 e β , where x > 0, α > 0, β > 0 is
A rv X with density f (x) = Γ(α)β αx
said to have a gamma distribution with parameter α and β.

We denote a rv with gamma distribution as X ∼ Gam(α, β).

Note: Gamma random variable: time required until r th occurrence.
Theorem 16.
Let X be a gamma rv with parameters α and β. Then
1. MX (t) = (1 − βt)−α , t < 1/β
2. E [X ] = αβ
3. V (X ) = αβ 2

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 Gamma Distribution Cont.

Example 17.
Prove the following
1. If Xi ∼ Γ(αi , β), i = 1, !2, · · · , n are independent, then
X n X n
Xi ∼ Γ αi , β
i=1 i=1
2. If X ∼ Γ(α, β), then kX ∼ Γ(α, kβ), where k > 0

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 Exponential Distribution

Definition 18.
A continuous random variable X is said to be an exponential random
variable with parameter β ifx its probability density function is of the
−
following form f (x) = β1 e β , x > 0, β > 0

Exponential distribution is a special case of Gamma distribution with

parameters α = 1 and β.
Note:Recall that in a Poisson process discrete events are being observed
over a continuous time interval. If we let W denote the time of the
occurrence of the first event, then W is a continuous rv. And, this W is
called exponential rv. That is, time required until first occurrence

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 Exponential Distribution Cont.

Theorem 19.
Consider a Poisson process with parameter λ. Let W denote the time of
the occurrence of the first event. W has an exponential distribution with
β = λ1

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 Exponential Distribution Cont.

Theorem 19.
Consider a Poisson process with parameter λ. Let W denote the time of
the occurrence of the first event. W has an exponential distribution with
β = λ1

Proof: The distribution function F for W is given by

F (w ) = P(W ≤ w ) = 1 − P(W > w )

The first occurrence of the event will take place after time w only if
no occurrences of the events are recorded in the time interval [0, w ].
Let X denote the number of occurrences of the event in this time
interval.
Thus, X Poisson rv with parameter λw

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 Exponential Distribution Cont.
Thus
P(W > w ) = P(X = 0) = e −λw
Now we can have f (w ) = λe −λw .
1
This is the pdf of an exponential rv with β = λ

Theorem 20.
n
X
If Xi ∼ Exp(β), and Xi are independent, then Xi ∼ Γ(n, β)
i=1

Example 21.
If the random variable X has a gamma distribution with parameters α = 1
and β = 2, then what is the probability density function of the random
variable Y = e X ?

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 Exponential Distribution Cont.
Thus
P(W > w ) = P(X = 0) = e −λw
Now we can have f (w ) = λe −λw .
1
This is the pdf of an exponential rv with β = λ

Theorem 20.
n
X
If Xi ∼ Exp(β), and Xi are independent, then Xi ∼ Γ(n, β)
i=1

Example 21.
If the random variable X has a gamma distribution with parameters α = 1
and β = 2, then what is the probability density function of the random
variable Y = e X ?
1√
ANS f (x) = 2x x
, x ≥1
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 Problems

Example 22.
Customers arrive at a certain shop according to an approximate Poisson
process at a mean frequency of 20 per hour. What is the probability that
the shopkeeper will have to wait for more than 5 minutes for the arrival of
the first customer?

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 Problems

Example 22.
Customers arrive at a certain shop according to an approximate Poisson
process at a mean frequency of 20 per hour. What is the probability that
the shopkeeper will have to wait for more than 5 minutes for the arrival of
the first customer?
−5
ANS: e 3

Example 23.
Telephone calls arrive at a college switchboard according to Poisson
process on an average of two every three minutes. what is the probability
that the waiting time is more than 2 minutes till the first call arrive after
10 AM?

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 Problems

Example 22.
Customers arrive at a certain shop according to an approximate Poisson
process at a mean frequency of 20 per hour. What is the probability that
the shopkeeper will have to wait for more than 5 minutes for the arrival of
the first customer?
−5
ANS: e 3

Example 23.
Telephone calls arrive at a college switchboard according to Poisson
process on an average of two every three minutes. what is the probability
that the waiting time is more than 2 minutes till the first call arrive after
10 AM?
4
ANS e − 3

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 Problems

Theorem 24 (Memoryless Property).

If X is an exponential random variable, P(X > x + h|X > h) = P(X > x)
for x > 0 and h > 0.
Proof: HW
Example 25.
According to the U.S. Geological Survey, earthquakes with magnitude at
least 7 occur on average 18 times a year (worldwide).
1. What is the probability that two consecutive such earthquakes are at
least 2 months apart?
2. What is the probability that there are no earthquakes in a 2-month
period?

Note: Solve this using Poisson and exponential

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 Problems

Theorem 24 (Memoryless Property).

If X is an exponential random variable, P(X > x + h|X > h) = P(X > x)
for x > 0 and h > 0.
Proof: HW
Example 25.
According to the U.S. Geological Survey, earthquakes with magnitude at
least 7 occur on average 18 times a year (worldwide).
1. What is the probability that two consecutive such earthquakes are at
least 2 months apart?
2. What is the probability that there are no earthquakes in a 2-month
period?

Note: Solve this using Poisson and exponential

ANS 0.05
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 Problems

Example 26.
Suppose that a system contains a certain type of component whose time,
in years, to failure is given by T . The random variable T is modeled nicely
by the exponential distribution with mean time to failure β = 5. If 5 of
these components are installed in different systems, what is the probability
that at least 2 are still functioning at the end of 8 years?

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 19 / 45
 Problems

Example 26.
Suppose that a system contains a certain type of component whose time,
in years, to failure is given by T . The random variable T is modeled nicely
by the exponential distribution with mean time to failure β = 5. If 5 of
these components are installed in different systems, what is the probability
that at least 2 are still functioning at the end of 8 years?

ANS 0.2627

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 19 / 45
 Problems

Example 26.
Suppose that a system contains a certain type of component whose time,
in years, to failure is given by T . The random variable T is modeled nicely
by the exponential distribution with mean time to failure β = 5. If 5 of
these components are installed in different systems, what is the probability
that at least 2 are still functioning at the end of 8 years?

ANS 0.2627
Example 27.
Let the life time of a radio in years, manufactured by a certain company
follows exponential distribution with average life 15 years. What is the
probability that, of eight such radios, at least two last more than 15 years.

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 19 / 45
 Problems

Example 28.
Suppose that telephone calls arriving at a particular switchboard follow a
Poisson process with an average of 5 calls coming per minute. What is the
probability that up to a minute will elapse until 2 calls have come in to the
switchboard?

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 20 / 45
 Problems

Example 28.
Suppose that telephone calls arriving at a particular switchboard follow a
Poisson process with an average of 5 calls coming per minute. What is the
probability that up to a minute will elapse until 2 calls have come in to the
switchboard?
ANS 0.96

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 Chi-Squared Distribution

Definition 29 (Chi-squared distribution).

A random variable X is said to have Chi-squared distribution with γ > 0
degrees of freedom, if X follows the Gamma distribution with parameter
α = γ2 and β = 2. We denote this variable by χ2γ .

The density function of Chi-squared random variable with degree of

freedom γ is given by

f (x) = 1
Γ(γ/2)2γ/2
x γ/2−1 e −x/2

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 Chi-Squared Distribution Conti.

Theorem 30.
1. The mgf of χ2γ : MX (t) = (1 − 2t)−γ/2 , t < 1/2
2. E [χ2γ ] = γ
3. Var (χ2γ ) = 2γ

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 Chi-Squared Distribution Conti.

Example 31.
If life length of certain kind of device follows chi-squared distribution with
dof 10. Find the probability that exactly 2 of 6 such devices in a system
will have to be replaced within the first 16 hours of operation. Assume
that life of the devices are independent.

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 Chi-Squared Distribution Conti.

Example 31.
If life length of certain kind of device follows chi-squared distribution with
dof 10. Find the probability that exactly 2 of 6 such devices in a system
will have to be replaced within the first 16 hours of operation. Assume
that life of the devices are independent.

Example 32.
If Xi , i = 1, 2, · · · , n are independent chi-squared rv with dof γi ,
Xn
i = 1, 2, · · · , n respectively, then show that Xi follows chi-squared with
i=1
n
X
dof γi
i=1

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 Normal Distribution

The normal distribution was introduced by the French mathematician

Abraham de Moivre in 1733 and was used by him to approximate
probabilities associated with binomial random variables when the binomial
parameter n is large.
Definition 33.
A random variable is said to be normally distributed with parameters µ
and σ 2 , and we write X ∼ N(µ, σ 2 ), if its density is

(x−µ)2
−
f (x) = √1 e 2σ 2 , −∞ < x, µ < ∞, σ > 0
2πσ

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 Normal Distribution Conti.

Definition 34.
Let X be normally distributed with parameters µ and σ.
σ2 t 2
1. The mgf is: MX (t) = e µt+ 2

2. E [X ] = µ
3. Var (X ) = σ 2

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 Normal Distribution Conti.

Definition 35 (Standard Normal Distribution).

The random variable following normal distribution with mean µ = 0 and
variance σ 2 = 1 is called standard normal random variable, usually it is
denoted by Z , i. e. X ∼ N(0, 1).

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 Normal Distribution Conti.

Definition 35 (Standard Normal Distribution).

The random variable following normal distribution with mean µ = 0 and
variance σ 2 = 1 is called standard normal random variable, usually it is
denoted by Z , i. e. X ∼ N(0, 1).

Theorem 36 (Standardization theorem).

Let X be normal with mean µ and standard deviation σ. The random
variable X σ−µ is standard normal.

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 Normal Distribution Conti.
Theorem 37.
 2
X −µ
If X ∼ N(µ, σ 2 ), then the rv σ ∼ χ21

Proof. HM
Theorem 38 (Normal probability rule).
Let X ∼ N(µ, σ 2 ). Then
1. P[−σ < X − µ < σ] = 0.68 or P[−1 < Z < 1] = 0.68
2. P[−2σ < X − µ < 2σ] = 0.95 or P[−2 < Z < 2] = 0.95
3. P[−3σ < X − µ < 3σ] = 0.997 or P[−3 < Z < 3] = 0.997

Example 39.
If X ∼ N(0, 1), what is the probability of the rv X less than or equal to
-1.72?

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 27 / 45
 Normal Distribution Conti.
Theorem 37.
 2
X −µ
If X ∼ N(µ, σ 2 ), then the rv σ ∼ χ21

Proof. HM
Theorem 38 (Normal probability rule).
Let X ∼ N(µ, σ 2 ). Then
1. P[−σ < X − µ < σ] = 0.68 or P[−1 < Z < 1] = 0.68
2. P[−2σ < X − µ < 2σ] = 0.95 or P[−2 < Z < 2] = 0.95
3. P[−3σ < X − µ < 3σ] = 0.997 or P[−3 < Z < 3] = 0.997

Example 39.
If X ∼ N(0, 1), what is the probability of the rv X less than or equal to
-1.72?
ANS 0.0427
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 Problems

Example 40.
If F (z) is a cdf for standard normal distribution, prove that
F (−z) = 1 − F (z)

Example 41.
If X ∼ N(3, 16), then what is P(4 ≤ X ≤ 8)?

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 Problems

Example 40.
If F (z) is a cdf for standard normal distribution, prove that
F (−z) = 1 − F (z)

Example 41.
If X ∼ N(3, 16), then what is P(4 ≤ X ≤ 8)?

ANS 0.2957
Example 42.
If X ∼ N(7, 4), what is P(15.364 ≤ (X − 7)2 ≤ 20.095)?

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 28 / 45
 Problems

Example 40.
If F (z) is a cdf for standard normal distribution, prove that
F (−z) = 1 − F (z)

Example 41.
If X ∼ N(3, 16), then what is P(4 ≤ X ≤ 8)?

ANS 0.2957
Example 42.
If X ∼ N(7, 4), what is P(15.364 ≤ (X − 7)2 ≤ 20.095)?

ANS 0.026

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 28 / 45
 Normal Approximation to Binomial

Theorem 43.
If X is a binomail rv with mean µ = np and variance σ 2 = npq, then the
limiting form of the distribution of Z = X√−np
npq , as n → ∞, is the standard
normal distribution, i. e. Z ∼ N(0, 1)

Continuity Correction: If X isbinomial rv and if we are looking for

P(X = 5), then by continuity correction we can approximate
P(4.5 < X < 5.5), i. e. if we seek the area under the normal curve to the
left of, say x, it is more accurate to use x + 0.5. This is a correction to
accommodate the fact, that a discrete distribution is being approximated
by a continuous distribution. The correction +0.5 is called a continuity
correction.

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 29 / 45
 Normal Approximation to Binomial Cont.

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 30 / 45
 Problems

Example 44.
A multiple-choice quiz has 200 questions each with 4 possible answers of
which only 1 is the correct answer. What is the probability that sheer
guesswork yields from 25 to 30 correct answers for 80 of the 200 problems
about which the student has no knowledge?

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 31 / 45
 Problems

Example 44.
A multiple-choice quiz has 200 questions each with 4 possible answers of
which only 1 is the correct answer. What is the probability that sheer
guesswork yields from 25 to 30 correct answers for 80 of the 200 problems
about which the student has no knowledge?

ANS 0.1196
Theorem 45.
If X has the distribution N(µ, σ 2 ) and if Y = aX + b, then
Y ∼ N(aµ + b, a2 σ 2 )

Proof: HW

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 31 / 45
 Problems

Example 46.
If the waist measurements X of 800 boys are normally distributed with
µ = 66cm and σ 2 25 cm, find the number of boys with waists greater than
or equal to 72cm

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 32 / 45
 Problems

Example 46.
If the waist measurements X of 800 boys are normally distributed with
µ = 66cm and σ 2 25 cm, find the number of boys with waists greater than
or equal to 72cm

ANS 92

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 32 / 45
 Problems

Example 46.
If the waist measurements X of 800 boys are normally distributed with
µ = 66cm and σ 2 25 cm, find the number of boys with waists greater than
or equal to 72cm

ANS 92
Example 47.
Suppose that a fuse has a life length X which may be considered as a
continuous random variable with an exponential distribution. There are
two processes by which the fuse may be manufactured. Process-I yields an
expected life length of 100 hours, while process-II yields an expected life
length of 150 hours. Suppose that process-II is twice as costly (per fuse)
as process-I, which costs C dollars per fuse. Assume, furthermore, that if a
fuse lasts less than 200 hours, a loss of K dollars is assessed against the
manufacturer. Which process should be used?
Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 32 / 45
 Problems

Example 48.
Suppose that the life length in hours, say T , of a certain electronic tube is
a random variable with exponential distribution with parameter β. A
machine using this tube costs C1 dollars/hour to run. While the machine
is functioning, a profit of C2 dollars/hour is realized. An operator must be
hired for a prearranged number of hours, say H, and he gets paid C3
dollars/hour. For what value of H is the expected profit greatest?

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 33 / 45
 Problems
Example 49.
The lifetimes of interactive computer chips produced by a certain
semiconductor manufacturer are normally distributed with mean 1.4 × 106
hours and standard deviation 3 × 105 hours. Use approximation to
approximate the probability that a batch of 100 chips will contain at least
80 whose lifetimes are less than 1.7 × 106 hours.

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 34 / 45
 Problems
Example 49.
The lifetimes of interactive computer chips produced by a certain
semiconductor manufacturer are normally distributed with mean 1.4 × 106
hours and standard deviation 3 × 105 hours. Use approximation to
approximate the probability that a batch of 100 chips will contain at least
80 whose lifetimes are less than 1.7 × 106 hours.
ANS 0.891

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 34 / 45
 Problems
Example 49.
The lifetimes of interactive computer chips produced by a certain
semiconductor manufacturer are normally distributed with mean 1.4 × 106
hours and standard deviation 3 × 105 hours. Use approximation to
approximate the probability that a batch of 100 chips will contain at least
80 whose lifetimes are less than 1.7 × 106 hours.
ANS 0.891
Example 50.
Laptop from a typical company has lifetime distribution such that it does
not fail until an external shock arrive (assume arrival of shock follows
Poisson process). The probability that laptop will have a lifetime more
than one year is 0.5. Find the probablity that the laptop will have life more
than 2 years given that it has already work for more than 1.5 years (correct
up to three decimal places)

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 34 / 45
 Problems
Example 49.
The lifetimes of interactive computer chips produced by a certain
semiconductor manufacturer are normally distributed with mean 1.4 × 106
hours and standard deviation 3 × 105 hours. Use approximation to
approximate the probability that a batch of 100 chips will contain at least
80 whose lifetimes are less than 1.7 × 106 hours.
ANS 0.891
Example 50.
Laptop from a typical company has lifetime distribution such that it does
not fail until an external shock arrive (assume arrival of shock follows
Poisson process). The probability that laptop will have a lifetime more
than one year is 0.5. Find the probablity that the laptop will have life more
than 2 years given that it has already work for more than 1.5 years (correct
up to three decimal places)

ANS 0.707
Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 34 / 45
 Problems

Example 51.
There are two types of batteries lifetime of type i battery is an exponential
random variable with parameter βi , i = 1, 2. The probability that a type i
battery from the bin is pi . If a randomly chosen battery is still operating
after t hours of use what is the probability it will still be operating after an
additional s hours?

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 35 / 45
 Problems

Example 51.
There are two types of batteries lifetime of type i battery is an exponential
random variable with parameter βi , i = 1, 2. The probability that a type i
battery from the bin is pi . If a randomly chosen battery is still operating
after t hours of use what is the probability it will still be operating after an
additional s hours?
−s −s
ANS p1 e β1 + p2 e β2
Example 52.
A bag of cookies is underweight if it weighs less than 500 pounds. The
filling process dispenses cookies with weight that follows the normal
distribution with mean 504 pounds and standard deviation 4 pounds. If
you select 5 bags randomly, what is the probability that exactly 2 of them
will be underweight?

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 35 / 45
 Problems

Example 51.
There are two types of batteries lifetime of type i battery is an exponential
random variable with parameter βi , i = 1, 2. The probability that a type i
battery from the bin is pi . If a randomly chosen battery is still operating
after t hours of use what is the probability it will still be operating after an
additional s hours?
−s −s
ANS p1 e β1 + p2 e β2
Example 52.
A bag of cookies is underweight if it weighs less than 500 pounds. The
filling process dispenses cookies with weight that follows the normal
distribution with mean 504 pounds and standard deviation 4 pounds. If
you select 5 bags randomly, what is the probability that exactly 2 of them
will be underweight?

ANS0.15173
Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 35 / 45
 Example 53.
Suppose that if you are s minutes early for an appointment, then you incur
the cost 6s and if you are s minutes late then you incur the cost 10s.
Suppose also that the travel time from where you presently are to the
location of your appointment is a continuous random variable having
uniform density over 60 to 80 minutes. Determine the time at which you
should depart if you want to minimize your expected cost.

Example 54.
Total number of calls received by the call center follows Poisson process
with rate 120 calls per hour. Find the probability that in a five working
day week, exactly in two of the five working days, the call center receives
his tenth call between first 6 to 8 min. of opening the call center each day.

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 36 / 45
 Example 53.
Suppose that if you are s minutes early for an appointment, then you incur
the cost 6s and if you are s minutes late then you incur the cost 10s.
Suppose also that the travel time from where you presently are to the
location of your appointment is a continuous random variable having
uniform density over 60 to 80 minutes. Determine the time at which you
should depart if you want to minimize your expected cost.

Example 54.
Total number of calls received by the call center follows Poisson process
with rate 120 calls per hour. Find the probability that in a five working
day week, exactly in two of the five working days, the call center receives
his tenth call between first 6 to 8 min. of opening the call center each day.

ANS0.2048

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 36 / 45
 Problems

Example 55.
A basketball team will play a 44-games season. Twenty six of these games
are against class A teams and 18 are against class B teams. Suppose that
the team will win each game against a class A team with probability 0.4
and will win each game against a class B team with probability 0.4.
Suppose also that the results of the different games are independent.
Approximate the probability that the basketball team wins total 20 games
in this 44-game season.

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 37 / 45
 Problems

Example 55.
A basketball team will play a 44-games season. Twenty six of these games
are against class A teams and 18 are against class B teams. Suppose that
the team will win each game against a class A team with probability 0.4
and will win each game against a class B team with probability 0.4.
Suppose also that the results of the different games are independent.
Approximate the probability that the basketball team wins total 20 games
in this 44-game season.

ANS0.0909

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 37 / 45
 Problems

Example 56.
Suppose four of you plan to go to Vasco railway station independent of
each other. You first walk to main gate, then walk to MES bus stop, after
that travel in a bus and finally walk to the destination. If the time spent in
each of the four stages of your travel are independent and exponentially
distributed with mean 20 minutes each, then find the probability that
exactly two out of four complete the journey within 35 minutes. Assume
that there is no waiting time between any of the four stages of trip.

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 38 / 45
 Problems

Example 56.
Suppose four of you plan to go to Vasco railway station independent of
each other. You first walk to main gate, then walk to MES bus stop, after
that travel in a bus and finally walk to the destination. If the time spent in
each of the four stages of your travel are independent and exponentially
distributed with mean 20 minutes each, then find the probability that
exactly two out of four complete the journey within 35 minutes. Assume
that there is no waiting time between any of the four stages of trip.

ANS 0.0486

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 38 / 45
 Problems

Example 57.
The IQ of a randomly selected individual is often supposed to follow a
normal distribution with mean 100 and standard deviation 15. Find the
probability that an individual has an IQ
(i) above 140 and
(ii) between 120 and 130, and
(iii) find a value x such that 99% of the population has IQ at least x.

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 39 / 45
 Problems

Example 58.
A power source gives an output voltage of 12 (volts). Because of random
fluctuations, the true voltage at any given time is V = 12 + X , where
X ∼ N(0, 0.1). The voltage is measured once an hour, and if it is outside
the interval [11.5, 12.5] the power source needs to be adjusted. What is
the probability that no adjustment is needed during a 24-hour period?

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 40 / 45
 Problems

Example 58.
A power source gives an output voltage of 12 (volts). Because of random
fluctuations, the true voltage at any given time is V = 12 + X , where
X ∼ N(0, 0.1). The voltage is measured once an hour, and if it is outside
the interval [11.5, 12.5] the power source needs to be adjusted. What is
the probability that no adjustment is needed during a 24-hour period?

Example 59.
Suppose that we are attempting to locate a target in three-dimensional
space, and that the three coordinate errors (in meters) of the point chosen
are independent normal random variables with mean 0 and standard
deviation 2. Find the probability that the distance between the point
chosen and the target exceeds 3 meters.

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 40 / 45
 Problems

Example 58.
A power source gives an output voltage of 12 (volts). Because of random
fluctuations, the true voltage at any given time is V = 12 + X , where
X ∼ N(0, 0.1). The voltage is measured once an hour, and if it is outside
the interval [11.5, 12.5] the power source needs to be adjusted. What is
the probability that no adjustment is needed during a 24-hour period?

Example 59.
Suppose that we are attempting to locate a target in three-dimensional
space, and that the three coordinate errors (in meters) of the point chosen
are independent normal random variables with mean 0 and standard
deviation 2. Find the probability that the distance between the point
chosen and the target exceeds 3 meters.

ANS 0.5222
Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 40 / 45
 Problems

Example 60.
The marks obtained by a number of students for a certain subject are
assumed to be approximately normally distributed with mean 65 and with
satandard deviation of 5. If 3 students are taken at random from this set,
what is the probability that exactly two of them hav emarks over 70?

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 41 / 45
 Problems

Example 60.
The marks obtained by a number of students for a certain subject are
assumed to be approximately normally distributed with mean 65 and with
satandard deviation of 5. If 3 students are taken at random from this set,
what is the probability that exactly two of them hav emarks over 70?

ANS 0.06357

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 41 / 45
 Problems

Example 60.
The marks obtained by a number of students for a certain subject are
assumed to be approximately normally distributed with mean 65 and with
satandard deviation of 5. If 3 students are taken at random from this set,
what is the probability that exactly two of them hav emarks over 70?

ANS 0.06357
Example 61.
The life time of electric bulbs has Gamma distribution with mean 20 and
standard deviation 10 seconds. A bulbs with life less than 20 seconds is
considered defective. What is the probability that of 6 randomly selected
bulbs at most two are defective?

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 41 / 45
 Problems

Example 60.
The marks obtained by a number of students for a certain subject are
assumed to be approximately normally distributed with mean 65 and with
satandard deviation of 5. If 3 students are taken at random from this set,
what is the probability that exactly two of them hav emarks over 70?

ANS 0.06357
Example 61.
The life time of electric bulbs has Gamma distribution with mean 20 and
standard deviation 10 seconds. A bulbs with life less than 20 seconds is
considered defective. What is the probability that of 6 randomly selected
bulbs at most two are defective?
ANS: 0.16579

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 41 / 45
 Problems

Example 62.
The annual rainfall (in inches) in a certain region is normally distributed
with µ = 40 and σ = 4. What is the probability that in two of the next
four years, the rainfall will exceed 45 inches? Also find the probability that
the total rainfall of first two years exceeds the total of the next two years.
Assume that the rainfall in different years are mutually independent.

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 42 / 45
 Problems

Example 62.
The annual rainfall (in inches) in a certain region is normally distributed
with µ = 40 and σ = 4. What is the probability that in two of the next
four years, the rainfall will exceed 45 inches? Also find the probability that
the total rainfall of first two years exceeds the total of the next two years.
Assume that the rainfall in different years are mutually independent.

Example 63.
A fire station is to be located on a road. Suppose that the road is of
infinite length stretching from point O outward to ∞. If the distance of a
fire from point O is exponentially distributed with mean 1 mile, then where
should the fire station be located so as to minimize the expected distance
from the fire?

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 42 / 45
 Problems

Example 62.
The annual rainfall (in inches) in a certain region is normally distributed
with µ = 40 and σ = 4. What is the probability that in two of the next
four years, the rainfall will exceed 45 inches? Also find the probability that
the total rainfall of first two years exceeds the total of the next two years.
Assume that the rainfall in different years are mutually independent.

Example 63.
A fire station is to be located on a road. Suppose that the road is of
infinite length stretching from point O outward to ∞. If the distance of a
fire from point O is exponentially distributed with mean 1 mile, then where
should the fire station be located so as to minimize the expected distance
from the fire?
ANS: ln 2 miles

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 42 / 45
 Conceptual

Assume that independent Bernoulli trials are being performed.

1. Random variable: number of occurrences of event A in a fixed number
of Bernoulli trials
Distribution: binomial
2. Random variable: number of Bernoulli trials required to obtain first
occurrence of A
Distribution: Geometric
3. Random variable: number of Bernoulli trials required to obtain rth
occurrence of A
Distribution: negative binomial

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 43 / 45
 Conceptual

Assume a Poisson process:

1. Random variable: number of occurrences of event A during a fixed
time interval
Distribution: Poisson
2. Random variable: time required until first occurrence of A
Distribution: exponential
3. Random variable: time required until rth occurrence of A
Distribution: gamma

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 44 / 45
 Thank you for your attention

Gauranga C Samanta (P. G. Dept. of Maths)Some Standard Continuous Distributions August 16, 2022 45 / 45