Problem 5E

There are times when a shifted exponential model is appropriate. That is, let the pdf of X be

(a) Define the cdf of X.

(b) Calculate the mean and variance of X.

Solution 5E

Step1 of 3:

We have A Exponential random variable X with pdf

We need to find,

(a) Define the cdf of X.

(b) Calculate the mean and variance of X.

Step2 of 3:

Let “X” be random variable which follows Exponential distribution with parameters .

That is X Exp()

The probability mass function of binomial distribution is given below

Where,

X = random variable

= parameter

= parameter

e = a constant it is approximately 2.7182.

a).

Consider,

To get a cdf of X we need to integrate above function with respect to x.

1).If X < then,

=

= 0.

2).If X then,

=

=

= [

=

=

=

Hence, the cdf is given by

Step3 of 3:

b).

Consider,

To get mean and variance of X we need to integrate above function with respect to x.

=

= []

=

=

= /

=

Hence, .

=

=

= -[0 - ]+

= + E(X)

= + []

= +

Hence, +

Therefore, the variance of X is given by:

V(X) = -

= + -

=

=

Hence, V(X) = .

Conclusion:

a).The cdf is given by

b).Variance