Question

Exercise 5.11. Suppose a fair 1-sided die is rolled, and the random variable X (s) outputs - 1 if the roll is 2, and 1 if the roll is 1,3, or 1. Calculate Mx(2).

Answer 1

A fair 4 sided die is rolled.

The random variable X(s) outputs -1 if the roll is 2; and 4 if the roll is 1, 3, or 4.

Now, when the 4 sided die is rolled, the possible outcomes are {1,2,3,4}.

So, there are 4 all possible cases.

Now, the outcomes are equally likely.

This means

$P(2)=\frac{1}{4}$

$P(1,3 or 4)=P(1)+P(3)+P(4)=\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}4$

This means,

X(s) takes value -1 with probability of 1/4.

X(s) takes value 4 with probability of 3/4.

Now, we have to find the value of

$M_{X}(z)$

This is nothing but the Moment Generating Function of X.

According to its definition, we know

$M_{X}(z)$

$=E(e^{xz})$

$=\sum{e^{xz}}P(X=x)$

Where, x is the value X takes and P(X=x) is the corresponding probability.

Ths becomes

$=e^{z(-1)}*\frac{1}{4}+e^{z(4)}*\frac{3}{4}$

$=\frac{3e^{4z}+e^{-z}}{4}$

Thus the answer is

The Moment Generating Function of X is

$M_{X}(z)=\frac{3e^{4z}+e^{-z}}{4}$