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.
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) = ,
X = random variable
e = a constant it is approximately 2.7182.
The moment generating function is given by
Let = y
If (1-) > 0
Substitute value in above equation we get
Simplifying above equation we get
Therefore,the moment generating function of X is .
The moment generating function of X is .