Question

Need a simple solution and please don't copy from other solutions.

2.3-11. If the moment-generating function of X is 2 find the mean, variance, and pmf of X

Answer 1

The moment generating function of the random variable X is defined as follows:

M(t) = E(e^tx )

For discrete random variable, we have

M(t) = E(e^tx ) = P(X = 0)e^0t + P(X = 1)e^1t + P(X = 2) e^2t + P(X = 3)e^3t+...

Comparing this mgf with the given , then we get the following probability distribution of X.

X	p(x)
1	2/5
2	1/5
3	2/5
Total	1

This is the pmf of X.

Let's find mean and variance of X.

Mean = E(X)

Therefore mean of X is 2

Variance of X = V(X)

V(X) = 〉 'p(z)(X-E(X))2 (2/5)(1-2)2+11/5)(2-2)2+(2/5)(3-2)

(2/5) + 0 + (2/5)-4/5-0.80

Therefore, variance of X is 0.80