Question

Let X and Y be independent Geometric(p) random variables.

(a) What is P(X < Y)?

(b) What is the probability mass function of the minimum min(X, Y )?

Answer #1

Let X and Y be independent Geometric(p) random variables. What
is P(X<Y)?

(14pts) Let X and Y be i.i.d. geometric random variables with
parameter (probability of success) p, 0 < p < 1. (a) (6pts)
Find P(X > Y ). (b) (8pts) Find P(X + Y = n) and P(X = k∣X + Y =
n), for n = 2, 3, ..., and k = 1, 2, ..., n − 1.

Let X1 and X2 be two independent geometric
random variables with the probability of success 0 < p < 1.
Find the joint probability mass function of (Y1,
Y2) with its support, where Y1 =
X1 + X2 and Y2 =
X2.

Let X and Y be geometric random variables with parameters 0.2
and 0.4. Find the Distribution of min(X,Y). Please show all
work.

Suppose X and Y are independent Geometric random variables, with
E(X)=4 and E(Y)=3/2.
a. Find the probability that X and Y are equal,
i.e., find P(X=Y).
b. Find the probability that X is strictly
larger than Y, i.e., find P(X>Y). [Hint: Unlike Problem 1b, we
do not have symmetry between X and Y here, so you must
calculate.]

Let X and Y be independent exponential random variables with
respective parameters 2 and 3.
a). Find the cdf and density of Z = X/Y .
b). Compute P(X < Y ).
c). Find the cdf and density of W = min{X,Y }.

. X,Y are absolutely continuous, independent random variables
such that P(X ≥ z) = P(Y ≥ z) = e−z for z ≥ 0. Find the expectation
of min(X,Y )

Let X and Y be a random variables with the joint probability
density function fX,Y (x, y) = { e −x−y , 0 < x, y < ∞ 0,
otherwise } . a. Let W = max(X, Y ) Compute the probability density
function of W. b. Let U = min(X, Y ) Compute the probability
density function of U. c. Compute the probability density function
of X + Y .

Let X and Y be independent discrete random variables with
pmf’s:
x
1
2
3
y
2
4
6
p(x)
0.2
0.2
0.6
p(y)
0.3
0.1
0.6
What is the probability that X + Y = 7

Let X and Y be random variables, P(X = −1) = P(X = 0) = P(X = 1)
= 1/3 and Y take the value 1 if X = 0 and 0 otherwise. Find the
covariance and check if random variables are independent.
How to check if they are independent since it does not mean that
if the covariance is zero then the variables must be
independent.

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