Jane chooses a number X at random from the set of numbers

{1, 2, 3, 4}, so that

P(X = k) =

1

4

for k = 1, 2, 3, 4.

She then chooses a number Y at random from the subset of

numbers {X, ...,

4

}; for example, if

X = 3, then Y is chosen at

random from {

3,

4}

.

(i) Find the joint probability distribution of X and Y and display

it in the form of a two-way table.

[5 marks]

(ii) Find the marginal probability distribution of Y , and hence

find E(Y ) and V ar(Y ).

[4 marks]

(iii) Show that Cov(X, Y ) = 5/8.

[4 marks]

(iv) Find the probability distribution of U = X + Y . [7