IMPACT COLLEGE OF ENGINEERING AND APPLIED SCIENCES,
Bangalore 560092
VISVESVARAYA TECHNOLOGICALUNIVERSITY
BELAGAVI-590018
MATHEMATICS for computer science-
BCS301
TOPIC :CONTINUOUS PROBABILITY DISTRIBUTION
EXPONENTIAL DISTRIBUTION
Presented by: NORMAL DISTRIBUTION
1IC23CD007-CHETHAN M N
1IC23CD008-DALAVAI VIJAY JAGAN
1IC23CD009-DHEERAJ JADHAV
Under the Guidanc of,
1IC23CD010-HEMANTH GOWDA S S
Mrs.Kavana C V
1IC23CD012-JAYANTH R
Department of basic science
1IC23CD013-MAHESH M
Continuous probability distribution
A discrete distribution is one in which the data can only take on certain
values, for example integers. A continuous distribution is one in which
data can take on any value within a specified range (which may be
infinite).
A continuous distribution is a statistical distribution wherein the possible
values of the random variable constitute a continuous range. This implies
that the variable can be any value within the specified range and not
necessarily restricted to the discrete individual value
Key Points
Continuous Random Variable: In a continuous probability distribution, the random
variable can take on any value within a specified interval or range. This means that
the variable can theoretically assume an infinite number of values within that range.
Probability Density Function (PDF): Persistent probabilities are frequently modeled
using a probability density function (PDF), which is a function that shows the relative
probability of the random variable obtaining different values across the domain of
interest. The area under the curve of PDF of a distribution between the needed
interval implies that a random variable has that value with the biggest probability.
Continuous Probability Distribution Formulas
1. Probability Density Function (PDF):
The probability density function f(x)describes the probability distribution of a continuous random variable X. It satisfies
the following properties:
f(x) ≥ 0 for al x in the range of X.
∫-∞∞ f(x)dx = 1 (total area under the curve equals 1).
2. Mean: Mean μ of a continuous random variable X is its average value, and it is given by:
μ = ∫-∞∞ x ⋅f(x)dx.
3. Variance: Variance σ 2 of a continuous random variable X measures the spread of its distribution, and it is given by: σ2
= ∫-∞∞ (x−μ)2⋅f(x)dx Standard deviation σ is the square root of the variance.
Note: The probability that a random variable X takes a value in the (open or closed) interval [a, b] is given by the integral
of a function. i.e. P(a ≤ X ≤b) = ∫ab f(x)dx
1) A random variable x has the density function 𝒑(𝒙)={ 𝒌𝒙𝟐, 𝟎≤ 𝒙≤ 𝟑
Numerical
𝟎, 𝒆𝒍𝒔𝒆𝒘𝒉𝒆𝒓𝒆 Evaluate k and find 𝒑( 𝒙≤ 𝟏),
𝒑(𝟏≤𝒙≤𝟐),𝒑(𝒙≤𝟐)𝒂𝒏𝒅 𝒑(𝒙>𝟏)
Exponential Distribution
In Probability theory and statistics, the exponential distribution is a continuous probability distribution that
often concerns the amount of time until some specific event happens. It is a process in which events happen
continuously and independently at a constant average rate. The exponential random variable can be either
more small values or fewer larger variables. For example, the amount of money spent by the customer on one
trip to the supermarket follows an exponential distribution.
Exponential Distribution Formula
The continuous random variable, say X is said to have an exponential distribution, if it has the following
probability density function: Where, λ is called the distribution rate.
Mean and Variance of Exponential Distribution
Mean:
The mean of the exponential distribution is calculated using the integration
by parts.
Variance:
To find the variance of the exponential distribution, we need to find the second moment of
the exponential distribution, and it is given by:
Now, substituting the value of mean and the second moment of the exponential
distribution, we get,
Numerical
1) At a certain city bus stop, three buses arrive per hour on an average. Assuming that the
time between successive arrivals is exponentially distributed, find the probability that the
time between the arrival of successive buses is (i) less than 10 minutes (i) at least 30
minutes.
Numerical
2)The sales per day for a shop is exponentialy distributed with
the average sale amounting to ₹100 and net profit is 8%. Find
the probability that the net profit exceeds ₹30 on two
consecutive days.
Normal Distribution
In a normal distribution, data is symmetrically distributed with no skew.
When plotted on a graph, the data follows a bell shape, with most values
clustering around a central region and tapering off as they go further
away from the center.
Normal distributions are also called Gaussian distributions or bell curves
because of their shape.
Normal distributions have key characteristics that are easy
to spot in graphs:
The mean, median and mode are exactly the same.
The distribution is symmetric about the mean— half the values fall
below the mean and half above the mean.
The distribution can be described by two values: the mean and the
standard deviation.
Formula of the normal curve
In a probability density function, the area under the curve
tells you probability. The normal distribution is a probability
distribution, so the total area under the curve is always 1 or
100%.
f(x) = probability
The formula for the normal probability density function looks
x = value of the variable
fairly complicated. But to use it, you only need to know the
μ = mean
population mean and standard deviation.
σ = standard deviation
For any value of x, you can plug in the mean and standard
σ2 = variance
deviation into the formula to find the probability density of
the variable taking on that value of x.
What is the standard normal distribution?
The standard normal distribution, also caled the z-distribution, is a
special normal distribution where the mean is 0 and the standard
deviation is 1. You only need to know the mean and standard
deviation of your distribution to find the z-score of a value. x = individual value
μ = mean
σ = standard deviation
Numerical 1
The life of a certain electric lamps is normally distributed with mean of 2040 hours and
standard deviation 60 hours. In a consignment of 2000 lamps, find how many would be
expected to burn for (i) more than 2150 hours (ii) less than 1950 hours and (iii) between
1920 hours and 2160 hours.
Numerical 2
The mean weight of 500 students at a certain school is 50 kgs and the standard deviation is 6 kgs. Assuming
that the weights are normally distributed, find the expected number of students weighing between (i)between
40 and 50 kgs (i) more than 60 kgs.
