ECSE-2500, Engineering Probability, Spring 2010, Rensselaer Polytechnic Institute
Homework 9, due 2pm Tues May 4 Fri May 7
- (20 pts) Let X be a Poisson random variable with
{$\alpha=\lambda=2$}. (Different books use {$\alpha$} and {$\lambda$} to
mean the same). Let Y be a binomial random variable with n=2 and p=0.5.
Let Z=X+Y.
- What is E[X]?
- What is E[Y]?
- What is fZ(z)? This may or may not simplify. If not, write the expression.
- What is E[Z]? Give a specific number, e.g., 42, with your reasoning.
- (55 pts) The experiment is to select a point uniformly in the
triangle (0,0), (1,0), (1,1). The random variables (X,Y) are the point's
coordinates.
- What is the marginal f(X)?
- What is the marginal f(Y)?
- What are E[X], E[Y], VAR[X], VAR[Y], COV[X,Y], {$\rho_{XY} $} ?
- Are X and Y independent? Why (not)?
- What is E[X|y= 0.5]?
- (25) Let X and Y be two normal N(0,1) random variables.
- What is the pdf of X+Y?
- What is the pdf of max(X,Y)?
- What is E[max(X,Y)]?
- What is E[min(X,Y)]?
- What is the joint pdf of (X+Y,X-Y)?
- (20) Let W, X, Y, Z be independent Raleigh random variables with
{$\alpha=1$} (defined on page 165).
- What is P[Z<min(W,X,Y)] ? The answer must be a number, with your reasoning.
- What is P[max(W,X)<min(Y,Z)] ? The answer must be a number, with your reasoning.