1) If X and Y are independent binomial random variables with identical parameters n and

p, calculate the conditional expected value of X given X+Y = m.

Hint: Begin by calculating the conditional mass function of X given that X+Y = m.

Use the fact that X+Y is a binomial distribution with parameters 2n and p.

2) (a) Find k if the joint probability density of random variables X and Y is given by

f(x,y) = ke^{-3x-4y} if x > 0 and y > 0 ;

0 elsewhere.

(b) Find the probability that the value of X falls between 0 and 1 while that of Y falls

between 0 and 2.

(c) Find the covariance for X and Y.

3) Given a pair of RV having joint density

f(x; y) = 1 if -y < x < y and 0 < y < 1;

0 otherwise

(a) show that the two marginal densities are

f_Y (y) = 2y if 0 < y < 1;

0 otherwise.

f_X(x) = 1 - x if 0 < x < 1;

= 1 + x if -1 < x <= 0;

0 otherwise.

and, hence, X and Y are not independent.

(b) Show that the two RV are uncorrelated. ie., Show that cov(X,Y) = 0.

4) Let a be a parameter > 0. Given a random sample of size n from the

population

f(x) = ax^{a-1} if 0 < x < 1;

0 otherwise.

find the maximum likelihood estimator for a.

5) An urn contains N coins among which theta of them are perfectly balanced, (the probabilities for heads and tails is 0.5), while the remaining N - theta coins are two headed. A coin is drawn at random, it is flipped twice, the result is noted, and the coin is returned to the urn without being otherwise inspected. If n repetitions of this experiment yielded 0,1,2 heads respectively, with frequencies n_0; n_1; and n_2, estimate theta using

(a) the method of moments,

(b) the method of maximum likelihood.

6) (Pearson curves) It was observed by Karl Pearson, one of the founders of modern statistics,

that the differential equation

f'(x)/f(x) = (d - x)/(a + bx + cx^2) ;

with certain prescribed boundary values, yields most of the distributions f(x) that are

important in statistics if appropriate values are chosen for the parameters a,b, c, and d.

Verify that this differential equation gives

(a) the normal distribution if b = c = 0 and a > 0:

(b) the exponential distribution if a = c = d = 0 and b > 0:

(c) the gamma distribution if a = c = 0, b > 0 and d > -b.

Subject | Mathematics |

Due By (Pacific Time) | 05/02/2015 12:00 am |

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