Question

(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.

Answer #1

(a) Let Y1,Y2,··· ,Yn be i.i.d.
with geometric distribution P(Y = y) = p(1−p)y-1 y=1, 2,
........, 0<p<1. Find a suﬃcient statistic for p.
(b) Let Y1,··· ,yn be a random sample of size n from
a beta distribution with parameters α = θ and β = 2. Find the
suﬃcient statistic for θ.

Let Y denote a geometric random variable with probability of
success p, (a) Show that for a positive integer a, P(Y > a) = (1
− p) a (b) Show that for positive integers a and b, P(Y > a +
b|Y > a) = P(Y > b) = (1 − p) b This is known as the
memoryless property of the geometric distribution.

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 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 independent Geometric(p) random variables. What
is P(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.)

X and Y ar i.i.d. exponential random variables
with mean = 2. Let Z = X + Y. The
probability that Z is less than or equal to 3 is:

Let {Xi} be i.i.d. random variables with P(Xi=−1) = P(Xi= 1) =
1/2. Let Sn= 1 +X1+. . .+Xn be symmetric simple random walk with
initial point S0 = 1. Find the probability that Sn eventually hits
the point 0.
Hint: Define the events A={Sn= 0 for some n} and for M >1, AM
= {Sn hits 0 before hitting M}.
Show that AM ↗ A.

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.

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.

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