This document discusses two continuous probability distributions: the uniform distribution and the exponential distribution. It provides the probability density functions and cumulative distribution functions for each distribution. Examples are also given to demonstrate calculating probabilities using each distribution.

1. Continuous Distributions
Uniform distribution.
Exponential distribution.
By one convention, a random variable X is
called continuous if its cumulative distribution
function is continuous.
Ex.
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2. Uniform distribution …..
 A random variable X is uniformly
distributed on the interval interval (a,b) if
its pdf is given by
 The cdf is

3. pdf and cdf of Uniform distribution
Probability density function Cumulative distribution function

4.  E(X)=(a+b)/2
 V(X)=(b-a)2/12

5. Example
 A bus arrives every 20 miutes at a
specified stop beginning at 6:40 AM
and continuing until 8:40 am. A
certain passenger does not know
the schedule, but arrives randomly
(uniformly distributed) between
7:00 am and 7:30 am every
morning. What is the probability
that the passenger waits more than
minutes for a bus?

6. Exponential distribution
 A random variable X is said to be
exponentially distributed with parameter
λ > 0 if its pdf is given by

7. Probability density function

8. Cumulative distribution function

9. Exponential distribution ….

10. Example
 suppose that the life of an industrial
lamp, in thousands of hours , it
exponentially distributed with failure
rate λ=1/3 (one of failure every 3000 hours,
on average).. The probability that the lamp
will last longer than its mean life of giving
3000 hours is given by P(X>3)=1-P(X<=3) =
1-F(3)
 P(X>3)=1-(1-e-3/3)=e-1 =0.368
Shashwat Shriparv
dwivedishashwat@gmail.com
InfinitySoft