Mast20004 probability - find mvt the moment generating

Question

Question 1. Let Y = X₁ + X₂ + .. . + X_N where N =^d Bi(n, p), X_i, =^d Bi(m, q) and N, X₁, X₂..... are independent.

(a) Find P_Y|N (z), the conditional probability generating function of Y given N , and state the values of z for which it is defined.

(b) Find P_Y(z), the probability generating function of Y, and state the values of z for which it is defined.

(c) Using PY(z), evaluate E(Y )

Question 2. Let X, Y , and Z be independent and uniformly distributed random variables on the interval (0, 1).

(a) Let U = X + Y . Find f_U(u), the probability density function of U, and state the values of u for which it is defined.

(b) Find M_U(t), the moment generating function of U, and state the values of t for which it is defined.

(c) Using M_U(t), evaluate E(U).

(d) Let V = X + Y + Z. Find f_V(v), the probability density function of V, and state the values of v for which it is defined.

(e) Find M_V(t), the moment generating function of V , and state the values of t for which it is defined.

(f) For i = 1, 2,. .., 1000, let X_i be uniformly distributed random variables on the interval (0, 1). Let W = X₁ + X₂ + .. . + X₁₀₀₀. Use the central limit theorem to approximate P (480 < W < 510).

Question 3. Let X and Y have joint probability density function f given by

                     2(x + y),     0 ≤ y ≤ x ≤ 1

f (x, y) =

                       0,              otherwise.

(a) Find fX|Y (x|y), the conditional density function of X given Y.

(b) Evaluate E(X|Y ), the conditional expectation of X given Y.

(c) Verify that E(X) = E(E(X|Y )).

Question 4. Let X =^d N (µ, σ²) and Y = e^X.

(a) Using the approximation formulae on Slides 367 and 368, derive expressions for E(Y ) and V (Y ) in terms of µ and o.

(b) Derive the exact expressions for E(Y ) and V (Y ) in terms of µ and σ.

(c) Let X =^d N (0, 1). Compare the approximated and exact values for E(Y ) and V (Y ).