1. A random variable X has uniform distribution on interval [a, b]. Show that Var(X) =1 /3(b^2 + ab + a^2) -1/4 (a + b)^2. You may want to use the fact Var(X) = E(X^2) - E(X)^2. 2. Suppose that the joint p.d.f. of two random variables X and Y is as follows: f(x; y) = 1/8c(x + y) for 0<= x <= 1 and 0 <= y <= 2 0 otherwise a) Find the value of constant c. b) Determine the conditional p.d.f of X for every given value of Y. c) P(X < 1/2│Y = 1/2) and E(X│Y = 1=2). d) Are X and Y independent? Why or why not? e) It turned out that E(X) = 5/9, E(X^2) = 7/18, E(Y ) = 11/9, and E(Y^2) = 16/9. Compute E(XY ). Given these, what is Var(2X - 3Y + 8)? ** Hint: for e) and question 2 c) in the below, use the fact V ar(Σi aiXi + c) = Σi a^2 V ar(Xi) +ΣijΣ aiajCov(Xi;Xj). Note that since aiaj = ajai, it can also be written asΣi a^2 V ar(Xi) + 2ΣΣi>jaiajCov(Xi;Xj), which implies V ar(aX +bY +c) = a^2V ar(X)+b^2V ar(Y )+2abCov(X; Y ) and V ar(aX +bY +cZ +d) = a^2V ar(X) + b^2V ar(Y ) + c^2V ar(Z) + 2abCov(X; Y ) + 2bcCov(Y;Z) + 2acCov(X;Z). -- At least there is a place, which I assure will always sing for you. It is my shy heart. --

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