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
;
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 ?
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.
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 and we
have
Challenging Let E1,...,En,... be independent, exponentially distributed random variables with
parameter λ > 0 and set
.
Calculate the limiting distribution of ZN for N →∞ by calculating the limiting cumulative distribution function
F(x) = lim P[Zn ≤ x].
N→∞
