Q.7.76 - A First Course in Probability

Question

Let $X$ be the value of the first die and $Y$the sum of the values when two dice are rolled. Compute the joint moment generating function of $X$ and $Y$.

Step-by-Step Solution

Answer

The joint moment generating functions of $X$ and $Y$ are $M_{x,y}\left(t_1,t_2\right)=\frac{1}{36}\sum _{x=1}^6\sum _{y=x+1}^{x+6}{e}^{t_{1}x+t_{2}y}$.

Step 1: Given Information

Let $X$ be the value of the first die and $Y$ the sum of the values when two dice are rolled. 

Step 2: Explanation

$Y=\left\{\begin{array}{l}2,3,4,5,6,7 if X=1\\ 3,4,5,6,7,8 if X=2\\ 4,5,6,7,8,9 if X=3\\ 5,6,7,8,9,10 if X=4\\ 6,7,8,9,10,11 if X=5\\ 7,8,9,10,11,12 if X=6\end{array}\right.$

$P(X=x,Y=y)=\frac{1}{36}$for all$\mathrm{x}$ and $\mathrm{y}$.

$M_{x,y}\left(t_1,t_2\right)=E\left[e^{t_{1}X+t_{2}Y}\right]$

Step 3: Explanation

$M_{x,y}\left(t_1,t_2\right)=E\left[e^{t_{1}X+t_{2}Y}\right]$

$=\sum _{x=1}^6\sum _{y=x+1}^{x+6}P(X=x,Y=y)e^{t_{1}x+t_{2}y}$

$=\frac{1}{36}\sum _{x=1}^6\sum _{y=x+1}^{x+6}{e}^{t_{1}x+t_{2}y}$

Step 4: Final Answer

The joint moment generating functions of  $X$ and $Y$ are $M_{x,y}\left(t_1,t_2\right)=\frac{1}{36}\sum _{x=1}^6\sum _{y=x+1}^{x+6}{e}^{t_{1}x+t_{2}y}$.