Question

If
? and ? are both identical but independent Geometric random
variables with parameter ?, find the probability that ?(? = ?).

Answer #1

(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.

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 {Xn} be a sequence of random variables that follow a
geometric distribution with parameter λ/n, where n > λ > 0.
Show that as n → ∞, Xn/n converges in distribution to an
exponential distribution with rate λ.

If X and Y are independent, where X is a geometric random
variable with parameter 3/4 and Y is a standard normal random
variable. Compute E(e X), E(e Y ) and E(e X+Y ).

If X and Y are independent, where X is a geometric random
variable with parameter 3/4 and Y is a standard normal random
variable. Compute E(e^X), E(e^Y ) and E(e^(X+Y) ).

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 )?

Let X be a geometric random variable with parameter p . Find the
probability that X≥10 . Express your answer in terms of p using
standard notation (click on the “STANDARD NOTATION" button
below.)

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

Suppose Y_1, Y_2,… Y_n denote a random sample of a geometric distribution with parameter p. Find the maximum likelihood estimator for p.

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