Question

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.

Answer #1

Let B~Binomial(n,p) denote a binomially distributed
random variable with n trials and probability of success
p. Show that B / n is a consistent estimator for
p.

(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 X be a discrete random
variable with positive integer outputs
a show that
p (X= K)= P( X> K-1) - P( X> k)
for any positive integer k
b Assume that for all k >I
we have P (X>k)=q^k use l()
to show that X is a geometric
random variable

let x be a discrete random variable with positive integer
outputs.
show that P(x=k) = P( x> k-1)- P( X>k) for any positive
integer k.
assume that for all k>=1 we have P(x>k)=q^k. use (a) to
show that x is a geometric random variable.

Let X denote a random variable that follows a binomial
distribution with parameters n=5, p=0.3, and Y denote a random
variable that has a Poisson distribution with parameter λ = 6.
Additionally, assume that X and Y are independent random
variables.
Derive the joint probability distribution function for X and Y.
Make sure to explain your steps.

How to use Chebyshev bound to achieve this question ?
Let X be a Geometric random variable, with success probability
p.
1) Use the Markov bound to find an upper bound for P (X ≥ a), for a
positive integer a.
2) If p = 0.1, use the Chebyshev bound to find an upper bound for
P(X ≤ 1). Compare it with the
actual value of P (X ≤ 1) which you can calculate using the PMF of
Geometric random...

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.
(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 denote a random variable that follows a binomial
distribution with parameters n=5, p=0.3, and Y denote a random
variable that has a Poisson distribution with parameter λ = 6.
Additionally, assume that X and Y are independent random
variables.
Using the joint pdf function of X and Y, set up the summation
/integration (whichever is relevant) that gives the expected value
for X, and COMPUTE its value.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 5 minutes ago

asked 5 minutes ago

asked 8 minutes ago

asked 10 minutes ago

asked 14 minutes ago

asked 14 minutes ago

asked 17 minutes ago

asked 18 minutes ago

asked 30 minutes ago

asked 47 minutes ago

asked 55 minutes ago

asked 1 hour ago