mth 524 2

MTH 524 Section D (Fall 2018) Homework 5

September 28, 2018

1. Consider a bivariate density, f, of a random vector (X,Y )), defined by f(x,y) = cx2y, 0 < x,y < 1, and = 0, for all other (x,y).

(a) Find the value of the constant c.

(b) Find P(X > Y ).

(c) Find the marginal densities of X and of Y and decide if X and Y are independent.

2. Consider a bivariate density, f, of a random vector (X,Y )), defined by f(x,y) = cx2y, 0 < y < x < 1, and = 0, for all other (x,y).

(a) Find the value of the constant c.

(b) Find P(X + Y > 1).

(c) Find the two conditional densities.

3. Suppose (X,Y,Z) is uniformly distributed in the solid sphere bounded by x2 + y2 + z2 = 1.

(a) Calculate the probability that the (X,Y,Z) lies in the cone z2 = x2 + y2.

(b) Find the marginal joint density of (X,Y ) and the marginal density fo Z.

4. Suppose X and Y are independent, Poisson random variables with parameters λ and µ respectively. Calculate the frequency function of X + Y .

5. Suppose X and Y are independent random variables, both uniformly distributed on the interval (0, 1). Calculate the cdfs and densities of (i) XY, and (ii) X/Y .

6. Suppose X and Y are continuous, independent random variables. Calculate the density fo X + Y when X is exponential with parameter λ and Y is gamma with parameters α = 2 and λ.

7. Suppose X and Y are independent, standard normal random variables.

(a) Calculate the joint density of (U,V ) where U = 3X + 4Y and V = 4X − 3Y .

(b) Decide if U and V are independent.

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