We say that two discrete random variables are independent if P[X = x, Y = y] = P

Question

We say that two discrete random variables are independent if P[X = x, Y = y] = P[X = x] × P[Y = y]for all specic values x and y that X and Y could be.

(a) The table below shows the distribution for two random variables X abd Y. For example,
P[X = 1, Y = 3] = 3/8.

Use the information in the table to show whether X and Y are independent, or not.

(b) For any two discrete random variables X and Y (not necessarily the ones in part (a)), show that
If X and Y are independent, then P[X = x|Y = y] = P[X = x], for any specic x and y that X
and Y could be.

Explanation / Answer

If independent P(X=1)P(Y=4)=1/8 P(X=1)P(Y=3)=3/8 P(X=2)P(y=4)=3/8 P(X=2)P(Y=3)=1/8 let P(X=1)=a then P(X=2)=1-a a P(Y=4)=1/8 and (1-a)P(Y=4)=3/8 P(Y=4)= 1/(8a)=3/(8-8a) so 8-8a=24a or 8=32a or a=1/4 and P(X=2)=3/4 P(X=1)P(Y=4)=1/8 or P(Y=4)=(1/8)(1/4)=1/32 and P(Y=3)=1-(1/32)=31/32 P(X=2)P(Y=3)=1/8 using above (3/4)(31/32)=93/128 So x and y are not independent. ----------------- If X and Y are independent, then P[X = x|Y = y] = P[X = x], for any speci?c x and y that X and Y could be. By definition of independent P(X=x|Y=y)=P(X=x, Y=y)/P(Y=y) and if independent = {P(X=x)P(Y=y)}/P(Y=y)=P(X=x)

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