Problem 7E

Find the moment-generating function for the gamma distribution with parameters α and θ.

Hint: In the integral representing E(etX ), change variables by letting y = (1 − θt)x/θ, where 1 − θt > 0.

Solution 7E

Step1 of 2:

We have A random variable X which follows gamma distribution with parameter

We need to find the moment generating function(mgf) of gamma distribution.

Step2 of 2:

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

That is X G()

The probability mass function of binomial distribution is given below

P(X) = ,

Where,

X = random variable

= parameter

= parameter

e = a constant it is approximately 2.7182.

Now,

The moment generating function is given by

=

=

Let = y

If (1-) > 0

=

=

Where,

Substitute value in above equation we get

=

Simplifying above equation we get

=

Therefore,the moment generating function of X is .

Conclusion:

The moment generating function of X is .