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Probability Theory Assignment 11 SOLVED

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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 ≤

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