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ECSE-2500, Engineering Probability, Spring 2010, Rensselaer Polytechnic Institute

Homework 9, due 2pm Tues May 4 Fri May 7

  1. (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.
    1. What is E[X]?
    2. What is E[Y]?
    3. What is fZ(z)? This may or may not simplify. If not, write the expression.
    4. What is E[Z]? Give a specific number, e.g., 42, with your reasoning.
  2. (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.
    1. What is the marginal f(X)?
    2. What is the marginal f(Y)?
    3. What are E[X], E[Y], VAR[X], VAR[Y], COV[X,Y], {$\rho_{XY} $} ?
    4. Are X and Y independent? Why (not)?
    5. What is E[X|y= 0.5]?
  3. (25) Let X and Y be two normal N(0,1) random variables.
    1. What is the pdf of X+Y?
    2. What is the pdf of max(X,Y)?
    3. What is E[max(X,Y)]?
    4. What is E[min(X,Y)]?
    5. What is the joint pdf of (X+Y,X-Y)?
  4. (20) Let W, X, Y, Z be independent Raleigh random variables with {$\alpha=1$} (defined on page 165).
    1. What is P[Z<min(W,X,Y)] ? The answer must be a number, with your reasoning.
    2. What is P[max(W,X)<min(Y,Z)] ? The answer must be a number, with your reasoning.