Question

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.)

Answer #1

Let X be a geometric random variable with parameter p .

Then the probability mass function of X is given as

Now,

= 1 - [ P(1) + P(2) + P(3) + P(4) + P(5) + P(6) + P(7) + P(8) + P(9) ]

Therefore,

(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.

Suppose that X|λ is an exponential random variable with
parameter λ and that λ|p is geometric with parameter p. Further
suppose that p is uniform between zero and one. Determine the pdf
for the random variable X and compute E(X).

Let X1,X2,...,Xn be a random sample from a geometric random
variable with parameter p. What is the density function ofU =
min({X1,X2,...,Xn})

Let X be a Poisson random variable with parameter λ and Y an
independent Bernoulli random variable with parameter p. Find the
probability mass function of X + Y .

Let X be an exponential random variable with parameter λ > 0.
Find the probabilities P( X > 2/ λ ) and P(| X − 1 /λ | < 2/
λ) .

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.

find the c.d.p(f(x)) of a random variable with geometric
distribution with parameter o<pci and based on the formula that
you derive for F(x) shows that it is increasing.

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
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variable. Compute E(e^X), E(e^Y ) and E(e^(X+Y) ).

a is a random variable which follows geometric distribution,
with parameter p = 0.1. b = ln(a)
1. what is the probability mass function of b.
2 Prob (b> ln(3)) = ?

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