MA 2631

Probability Theory

Assignment 11 – last assignment

based on Lectures of Chapter 6.1–6.2

1. Let X, Y be two random variables with joint cdf FX,Y and marginal cdfs FX, FY . For x,

y ∈ R, express

P[X > x; Y ≤ y]

in terms of FX,Y and FX, FY

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 pX,Y of the random variable X and

Y is given by

pX,Y (1, 1) =pX,Y (1, 2) = pX,Y (1, 3) = 1

12

;

pX,Y (2, 1) =pX,Y (2, 2) = pX,Y (2, 3) = 1

4

.

a) Calculate the marginal probability mass distributions pX and pY .

b) Are X and Y independent?

c) What is the probability mass distribution of the random variable Z =

X

Y

?

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

fX,Y (x, y) =

cxy if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1;

0 else,

for some constant c.

a) What is the value of c?

b) Are X and Y independent?

c) Calculate E[X].

6. Let X1, . . . , Xn 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, . . . Xn}, Z = max{X1, X2, . . . Xn}

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 X ∼ N

µX, σ2

X

and Y ∼ N

µY , σ2

Y

we

have

X + Y ∼ N

µX + µY , σ2

X + σ

2

Y

Challenging Let E1, . . . , En, . . . be independent, exponentially distributed random variables with

parameter λ > 0 and set

Zn = max

{1≤n≤N}

En −

log N

λ

.

Calculate the limiting distribution of ZN for N → ∞ by calculating the limiting

cumulative distribution function

F(x) = lim

N→∞

P[Zn ≤

MA 2631

# Probability Theory Assignment 11 SOLVED

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