statistics

Statistical Inference I: J. Lee Assignment 6

Problem 1. SELECT ONE OF THE FOLLOWING THREE PROBLEMS TO DO AND TURN IN. You should understand how to do all three, since they provide good practice.

(a) Let X and Y have joint density

f(x, y) =

{ c(y2 −x2)e−y if −y ≤ x ≤ y, 0 < y < ∞ 0 otherwise.

(i) Find c (you may leave unevaluated, if you write it in explicit terms involving an integral).

(ii) Compute the marginal densities of X and of Y (be explicit about all cases!).

(iii) Compute P (Y > 2X).

(iv) Compute E(X).

(v) Are X and Y independent? JUSTIFY YOUR ANSWER.

(b) Let X and Y have joint density

f(x, y) =

{ cxy2 if 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 0 otherwise.

(i) Find c (you may leave unevaluated, if you write it in explicit terms involving an integral).

(ii) Compute the marginal densities of X and of Y (be explicit about all cases!).

(iii) Compute P (Y > 2X).

(iv) Compute P (|Y −X| < 0.5). (v) Are X and Y independent? JUSTIFY YOUR ANSWER.

(c) Let X and Y have joint density

f(x, y) =

{ cxy2 if 0 ≤ x, 0 ≤ y, x + y ≤ 1 0 otherwise.

(i) Find c (you may leave unevaluated, if you write it in explicit terms involving an integral).

(ii) Compute the marginal densities of X and of Y (be explicit about all cases!).

(iii) Compute Cov(X, Y )

(iv) Compute var(X2 + Y ).

(v) Are X and Y independent? JUSTIFY YOUR ANSWER.

Problem 2. If E(3X) = var(X/2) and var(2X) = 3, find (a). E [ (2 + X)2

] and (b). var(4 + 3X).

Problem 3. The random variables X and Y have a joint density function given by

f(x, y) =

 

2e−2x

x if 0 ≤ x < ∞, 0 ≤ y ≤ x

0 otherwise.

Compute cov(2X, Y + 3).

1