Question

Let X and Y have the following joint probability distribution
x/y
1
3
5

2
1/8
1/24
1/12

4
1/4
1/4
0

6
1/8
1/24
1/12

Find the probability distribution of Y
Find the conditional distribution of Y given X = 2
Examine whether X and Y independent

Answer 1

$X$ and $Y$ have the joint probability distribution (Table).

1) is the probability distribution of $Y$($P\{Y=1\}$ equals sum of values of the first row of Table, $P\{Y=2\}$ equals sum of values of the second row of Table, $P\{Y=3\}$ equals sum of values of the third row of Table).

2) is the conditional distribution of $Y$ given $X = 2$$(P\{Y=1|X=2\}, P\{Y=2|X=2\}, P\{Y=3|X=2\};\\ P\{Y=t|X=2\}=\frac{P\{X=2, Y=t\}}{P\{X=2\}}).$.

3) Table 3 shows that $X$ and $Y$ are not independent.

From Table 3 we see that $P\{X=x,Y=y\}\neq P\{X=x\}P\{Y=y\}.$

Values of Table 3 are all possible products of values of Tables 1) and 4) where

4) is the probability distribution of $X$($P\{X=k\}$ equals sum of values of the k_th column of Table).