Question

The independent variables X and Y have probability distributions


`P(X=x) = (1)/(5) , x = 1, 2, 3, 4, 5\\P(Y=y) = 1/y , y = 2, 3, 6`

Find
`P(Y\ \textgreater \ X)`

Answer 1

`Y>X` for the following
`(x,y)`:

(1, 2), (1, 3), (1, 6)

(2, 3), (2, 6)

(3, 6)

(4, 6)

(5, 6)

So we have

`P(Y>X)=P(X=1,Y=2)+P(X=1,Y=3)+\cdots+P(X=5,Y=6)`

`X,Y` are independent, so the joint probabilities are

`P(X=x,Y=y)=P(X=x)\cdot P(Y=y)=\frac1{5y}`

Then

`P(Y>X)=\frac1{10}+\frac2{15}+\frac5{30}=\frac25`