Q. 7.48 - A First Course in Probability

Question

If $Y=a X+b$ where a and b are constants, express the moment generating function of $Y$ in terms of the moment generating function of $X$. 

Step-by-Step Solution

Answer

The $e^{tb}{M}_X\left(at\right)$ moment generating function and in the fourth equality we used linearity of expectation.

Step 1: Given Information

We need to find the moment generating function of $Y=a X+b$.

Step 2: Explanation

We need to find the moment generating function of $Y=a X+b$where $a,b\in \mathrm{ℝ}$.

We evaluate to get

$M_Y\left(t\right)=E\left[e^{tY}\right]$

Substitute,

$=E\left[e^{t(aX+b)}\right]$.

Step 3: Explanation

Simplify, 

$=E\left[e^{atX}{e}^{tb}\right]$

$=e^{tb}E\left[e^{atX}\right]$

Evaluate that, 

$=e^{tb}{M}_X\left(at\right)$

In first and last equality we used the definition of the moment generating function and in the fourth equality we used linearity of expectation.

Step 4: Final answer

The $e^{tb}{M}_X\left(at\right)$ moment generating function and in the fourth equality we used linearity of expectation.