statistics

Statistical Inference I: J. Lee Assignment 7 Due on Nov 17, 2015

Problem 1. Let X and Y be continuous random variables with joint probability density function given by

f(x, y) =

{ Cx2 if x ≥ 0, x < 4, x ≥−y, and x ≥ y/2 0 otherwise.

(a) The marginal density, fY (y), of Y . (Be explicit about all cases.)

(b) The conditional density, fX|Y (x|2), of X given Y = 2. Be explicit about all cases!

(c) P (X > 3 | Y = 2).

(d) E(X | Y = 2).

(e) P (X > 3 | Y ≥ 2).

Problem 2. First a point Y is selected according to the density

fY (y) =

{ y if 0 < y <

√ 2

0 otherwise.

Then another point X is selected (uniformly) at random from the interval (−Y, Y ).

(a) Find the joint probability density function for X and Y .

(b) Find the (marginal) probability density function of X.

(c) Compute P (X > 0).

Problem 3. Consider the maze shown below. There are three cells (Cell 1, Cell 2, and Cell 3) and two deadly (quite permanent) outcomes (Death By Poison, and the dreaded Death By Guillotine). A rat is initially placed in cell 1. When the rat enters Cell i, he wanders around within the cell for Xi minutes, where Xi is uniformly distributed between 0 and i, and then he exits the cell by picking one of the doors at random (e.g., if there are 3 doors, he picks each with probability 1/3).

(a) Find the probability that the rat dies by poison. (Recall that he starts in Cell 1.)

(b) What is the expected number of minutes that the rat lives?

(c) What is the probability the rat visits Cell 3 before he dies?

(In each of the above problems, you should DEFINE a set of unknowns precisely, set up a system of equations involving the unknowns, and then solve for the final answer (time permitting). You need not solve the actual system, if you give a complete description of how you would finish solving the problem.)

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