Solution: (1) Because the random variables X and Y are independently and identically distributed.

The possible values of u are 1, 2; The possible values of v are 1, 2.

p(U = 1； v = 1)= P(X = 1； Y = 1)= P(X = 1)* P(Y = 1)= 2/3 * 2/3 = 4/9

p(U = 1； V=2)=0

p(U = 2； v = 1)= P(X = 2； y = 1)+P(X = 1； y = 2)= 1/3 * 2/3+ 1/3 * 2/3 = 2/9+2/9 = 4/9

p(U = 2； v = 2)= P(X = 2； y = 2)= 1/3 * 1/3 = 1/9

Draw the probability distribution of (u, v) again, which is omitted here.

(2) Do you want to ask cov(U, v)?

EU = 1 * 4/9+2 * 5/9 = 14/9

EV = 1 * 8/9+2 * 1/9 = 10/9

Euv =1* 4/9+2 * 4/9+4 *1/9 =16/9 (UV distribution omitted).

cov(U，V)= EUV-EUEV = 16/9- 14/9 * 10/9 = 4/8 1

Ask me if you don't understand.