CONTINUOUS PROBABILITY
DISTRIBUTION(NORMAL
DISTRIBUTION)
CHAPTER OUTLINE
Continuous Probability Distributions
The Normal Probability Distribution
Approximating the Binomial Distribution
by Using the Normal Distribution
Approximating the Poisson Distribution by
Using the Normal Distribution
Continuous Probability Distributions
If a random variable is a continuous variable, its
probability distribution is called a continuous probability
distribution.
A continuous probability distribution differs from a discrete
probability distribution in several ways.
The probability that a continuous random variable will
assume a particular value is zero.
As a result, a continuous probability distribution cannot
be expressed in tabular form.
Instead, an equation or formula is used to describe a
continuous probability distribution.
Most often, the equation used to describe a
continuous probability distribution is called a
probability density function. Sometimes, it is
referred to as a density function, a PDF, or a
pdf.
THE NORMAL DISTRIBUTION
The normal distribution is a probability
density function for a continuous
variable, and is represented by a
continuous curve.
Most widely used continuous
distribution
It has wide range of practical
application like where the r.v. are
human characteristics such as height,
weight, years of life etc.
Area under curve is equal to the sum of
expected frequencies
freq
Some variable
Cannot evaluate the probability of the
variable being exactly equal to some value
(that area of the curve is soooo small)
Must estimate the frequency of observations
falling between two limits
Won’t work
freq
2 3
estimate the frequency of observations
falling between 2 and 2.2
freq
2 3
Area under the curve = probability
Area under whole curve = 1
Probability of getting specific number is 0,
e.g. P(X=2) = 0
CHARACTERISTICS OF
NORMAL DISTRIBUTION
Symmetric, bell-shaped curve.
Shape of curve depends on population mean
and standard deviation .
Center of distribution is .
Spread is determined by . Larger the s.d.,
wider, flatter curves exist, exhibiting more
variability in the data.
Mean = Median = Mode
Two tails of the normal curve extend
indefinitely and never touch the horizontal
axis
Total area under the normal curve is 1
Learning Objective
6-3: Describe the
THE POSITION AND SHAPE
properties of the
normal distribution
and use a
OF THE NORMAL CURVE (1
cumulative normal
table. OF 2)
a) Two normal curves with different means and
equal standard deviations. If 1 is greater
than 2, the normal curve with mean 1 is
centered farther to the right.
Copyright ©2017 McGraw-Hill Education. All rights reserved. 6-11
Learning Objective
6-3: Describe the
properties of the
normal distribution
THE POSITION AND SHAPE OF
THE NORMAL CURVE (2 OF 2)
and use a
cumulative normal
table.
b) Two normal curves with same mean and
different standard deviations. If 1 is greater
than 2, the normal curve with standard
deviation 1 is flatter and more spread out.
Copyright ©2017 McGraw-Hill Education. All rights reserved. 6-12
Learning Objective
6-3: Describe the
properties of the
normal distribution
and use a
cumulative normal
table. THREE IMPORTANT PERCENTAGES
Copyright ©2017 McGraw-Hill Education. All rights reserved. 6-13
Learning Objective
6-3: Describe the
properties of the
normal distribution
and use a
FIGURE 6.7 FINDING NORMAL
cumulative normal
table. CURVE AREAS
x μ
z
σ
Copyright ©2017 McGraw-Hill Education. All rights reserved. 6-14
Learning Objective
6-4: Use the
normal distribution
to compute FINDING NORMAL
probabilities
PROBABILITIES
1. Formulate the problem in terms of x values
2. Calculate the corresponding z values, and
restate the problem in terms of these z
values
3. Find the required areas under the standard
normal curve by using the table
Note: It is always useful to draw a picture
showing the required areas before using
the normal table
Copyright ©2017 McGraw-Hill Education. All rights reserved. 6-15
PROBABILITY ABOVE 75?
Probability student scores higher than 75?
0.08
0.07
0.06
0.05
Density
0.04 P(X > 75)
0.03
0.02
0.01
0.00
55 60 65 70 75 80 85
Grades
PROBABILITY = AREA UNDER CURVE
Calculus?! You’re kidding, right?
But there are an infinite number of normal
distributions (one for each and )!!
Solution is to “standardize.”
STANDARDIZING
Take value X and subtract its mean from it,
and then divide by its standard deviation .
Call the resulting value Z.
That is, Z = (X- )/
Z is called the standard normal. Its mean
is 0 and standard deviation is 1.
Then, use probability table for Z.
How to determine what proportion of a
normal population lies above/below a
certain level
If distribution of Hobbit
heights is normal with
mean = 120 cm, SD = 20
120 cm
Half < 120 & half >120
What is probability of
finding a Hobbit taller
than 130 cm??
The
average
Hobbit
Calculate the normal deviate
- Any point on normal curve
- Here, 130 cm Mean
Xi -
Z=
- Normal deviate SD
- Test statistic
Z = (130-120)/20 = 0.5
Table B.2; Zar
Table A S & R
P (probability) (Xi >130 cm) = P (Z>0.50) =
0.3085 or 30.85%
EXAMPLE
A workshop produces 2000 units per day. The
average weight of units is 130 kg with the
standard deviation of 10 kg. Assuming
normal distribution, how many units are
expected to weigh less than 142 kg?
EXAMPLE
Time taken by the crew of a company to
construct a small bridge is a normal variate
with mean 400 labour hours and standard
deviation 200 labour hours.
What is the prob. That the bridge get
constructed between 350 to 450 labour
hours?
Find the prob. That the company takes at
most 500 hours to complete the flyover.
EXAMPLES
Delhi Traffic Police claims that whenever any rally is
organized in the city, traffic in the city is seriously disrupted.
On the day of rally, city’s traffic is disrupted for about 3
hours on an average with a S.D. of 45 minutes. It is believed
that the distribution of traffic is normally distributed. If on a
certain day, a rally is organized in the city what is the prob.
That:
Traffic was disrupted upto 2 hours
Traffic was disrupted upto 5 hours
Traffic remained disrupted between 1 to 4 hours
The mean inside diameter of a sample of 500 washers
produced by a machine is 5.02 mm, and the S.D. is 0.05mm.
The purpose for which these washers are intended, allows a
maximum tolerance in the diameter of 4.96 to 5.08 mm,
otherwise the washers are considered defective. Determine
the percentage of defective washers produced by the
machine assuming the diameters are normally distributed.
EXAMPLE
A placement company has conducted a written test to
recruit people in a software company. Assume that the
test marks are normally distributed with mean 120
and standard deviation 50. Calculate the following :
Prob. Of randomly obtaining scores greater than
200 in the test
Prob. Of randomly obtaining a score that is 180 or
less
Prob. Of randomly obtaining a score less than 80
Prob. Of randomly selecting a score between 70 to
170 for the exam
Prob. Of randomly obtaining a score between 80 to
110
EXAMPLE
New improved variety of small cars
manufactured in India by Maruti-Suzuki
Automobiles Limited, are recommended for
servicing after every 10,000 kms. On an
average 2 hours are required for servicing
each such car with a S.D. of 0.5 hour.
The service time is assumed to be normally
distributed, calculate the probability that a
car will take more than 3 hrs; less than 1.2
hrs.
EXAMPLE
A company manufacturers detergents in
packs of 500 gms. The quantity of detergents
packed follows a normal distribution with
mean 500 gms, and S.D. 2 gms. In a lot of
1000 packets, find the
(i) no. of packets with weight exceeding
501 gms.
(ii) no. of packets with weight less than
498 gms.
EXAMPLE
In a distribution exactly normal 7% of the
items are under 35 and 89% are under 63.
What are the mean and standard deviation of
the distribution.
EXAMPLE
Extensive testing indicates that the lifetime of
the Everlast automobile battery is normally
distributed with mean of µ=60 months and a
standard deviation of 6 months. The
manufacturer has decided to offer a free
replacement battery to any purchaser whose
everlast battery does not last at least as long as
the minimum lifetime specified in its guarantee.
How can the manufacturer establish the
guarantee perios so that only 1 percent of the
batteries will need to be replaced free of charge?
NORMAL APPROXIMATION OF
BINOMIAL PROBABILITIES
In cases where no. of trials are greater than
20, np ≥ 5 and n(1-p)≥ 5, the normal prob.
Distr. Can be used as an approximation of
binomial probabilities.
For using the normal approximation of the
binomial probabilities, we will convert two
parameters of the B.D. n and p, to two
parameters of the N.D. μ and σ.
EXAMPLE
A fast food owner calculates that about
40% of his customers order soft drink
with burgers. If 500 customers visit the
fast food joint, what is the probability that
more than 190 will order soft drinks with
burgers?
EXAMPLE
A machine produces bolts of a certain type.
In a sample of 100 bolts, if the number of
defectives is less than 12, the entire
production lot is accepted. Find the
probability that the lot is accepted when the
machine produces 20% defective bolts.
EXAMPLE
A retail appliance store owner knows from
experience that the no. of LCD television sets
he sells each month is a r.v. having
approximately a N.D. with the mean 32.3 and
the SD 4.2. calculate the probability that in
any given month he will sell
Exactly 25 LCD television sets
At most 25 LCD television sets
APPROXIMATION TO POISSON
DISTRIBUTION
A poisson distribution is approximated by a
normal distribution under the condition that
mean should be at least equal to 5
EXAMPLE
The average number of accidents in a large
factory is 8 per quarter. The occurrence of
accidents follows poisson distribution.
Calculate the probability that in a given
quarter, at least 10 accidents would occur.
Calculate the probability that during a six-
month period, exactly 12 accidents would
occur.
