[SOLVED] MA2631 Assignment 11

1. Let X, Y be two random variables with joint cdf F_X,Y and marginal cdfs F_X, F_Y . For x, y ∈R, express

P[X > x;Y ≤ y]

in terms of F_X,Y and F_X, F_Y

2. Assume that there are 12 balls in an urn, 3 of them red, 4 white and 5 blue. Assume that you draw 2 balls of them, replacing any drawn ball by a ball of the same color. Denote by X the number of drawn red balls and by Y the number of drawn white balls. Calculate the joint probability mass distribution of X and Y as well as the marginal distributions. Are X and Y independent?
3. Assume that the joint probability mass distribution p_X,Y of the random variable X and Y is given by

;

1. Calculate the marginal probability mass distributions p_X and p_Y .
2. Are X and Y independent?
3. What is the probability mass distribution of the random variable?

2

4. et X and Y be two independent standard-normal distributed random variables and define Z = X² + Y ². Calculate the cumulative distribution function of Z. Which distribution follows Z?
5. Let X, Y be two jointly distributed random variables with joint density

if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1;

X,Y (x,y) =

0       else,

for some constant c.

1. What is the value of c?
2. Are X and Y independent?
3. Calculate E[X].

6. Let X₁,…,X_n be independent and identically distributed random variables with density f and cumulative distribution function F. Calculate density and cumulative distribution function of

Y = min{X₁,X₂,…X_n},                    Z = max{X₁,X₂,…X_n}

in terms of f and F.

8 points per problems

Additional practice problems (purely voluntary – no points, no credit, no grading):

Standard Carlton and Devore, Section 4.1: Exercises 1, 3, 4, 8, 11, 13, 14, 19 ; Section 4.2: Exercises 23, 24, 29

Hard Prove that for independent random variables  and we

have

Challenging Let E₁,…,E_n,… be independent, exponentially distributed random variables with

parameter λ > 0 and set

.

Calculate the limiting distribution of Z_N for N →∞ by calculating the limiting cumulative distribution function

F(x) = lim P[Z_n ≤ x].

N→∞