Q. 6.39 - A First Course in Probability

Question

Two dice are rolled. Let X and Y denote, respectively, the largest and smallest values obtained. Compute the conditional mass function of Y given X = i, for i = $1,2, ... ,6$. Are X and Y independent? Why? 

Step-by-Step Solution

Answer

The conditional mass function of Y and state X and Y are not independent.

Step 1: Given information

Two dice are rolled. consider X and Y denote the two events that represent the smallest and largest values respectively.

$X=i for i=1,2, ......, 6.$

Step 2: Explanation

Calculate the conditional mass function of Y given X = i for $i=1,2,3,4,5,6$ as follows:

Here X is the larger of the rolls and $X=1$ then there are two cases.

Consider R1 and R2 to be the events that represent the first roll and second roll.

Case $1$: $For j<i$

$$\begin{gathered} P\left(Y=j|X=i\right) =\frac{P(Y=j,X=i)}{P(X=i)} \\ =\frac{P\left\{\left(R_1=j,R_2=i\right)\cup \left(R_1=i,R_2=j\right)\right\}}{P(X=i)} \\ =\frac{2}{36\times P\left(X=i\right)} \end{gathered}$$

Case $2$: For i = j

$$\begin{gathered} P\left(Y=i|X=j\right) =\frac{P(Y=i,X=i)}{P(X=i)} \\ =\frac{1}{36\times P\left(X=i\right)} \end{gathered}$$

Therefore to calculate P (X = i) as follows:

$$\begin{gathered} P(X=i)=P\left\{\left[\underset{k=1}{\overset{i-1}{\cup }}\left(R_1=k,R_2=i\right)\cup \left(R_1=i,R_2=k\right)\right]\cup \left(R_1=i,R_2=i\right)\right\} \\ =\sum _{k=1}^{i-1}\left(\frac{1}{36}+\frac{1}{36}\right)+\frac{1}{36} \\ =\frac{2i-1}{36} \end{gathered}$$

Hence, the conditional mass function of Y given X = i for as

$P(Y=j\mid X=i)=\left\{\begin{array}{cl}\frac{2}{2i-1}& j<i\\ \frac{1}{2i-1}& j=i\end{array}\right.$

Check whether X and Y are independent or not.

If two random variables are independent, then $Y\le X$.

Here, $P\left(Y=j | X=i\right)=P\left(Y=j\right)$

Hence $P\left(Y=j | X=j\right)$ depends on i then X and Y are not independent