Question

1. Show that if X is a Poisson random variable with parameter
λ, then its variance is λ

2.Show that if X is a Binomial random variable with parameters
n and p, then the its variance is npq.

Answer #1

A Poisson random variable is a variable X that takes on the
integer values 0 , 1 , 2 , … with a probability mass function given
by p ( i ) = P { X = i } = e − λ λ i i ! for i = 0 , 1 , 2 … ,
where the parameter λ > 0 .
A)Show that ∑ i p ( i ) = 1.
B) Show that the Poisson random...

Write a proof to show that if X is a Binomial random variable
with parameters n and p, then its variance is
npq.

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 a Poisson random variable with parameter λ > 0.
Determine a value of λk that maximizes P(X = k) for k ≥
1.

Let X and Y be independent random variables following Poisson
distributions, each with parameter λ = 1. Show that the
distribution of Z = X + Y is Poisson with parameter λ = 2. using
convolution formula

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).

5.2.12. Let the random variable Zn have a Poisson distribution
with parameter μ = n. Show that the limiting distribution of the
random variable Yn =(Zn−n)/√n is normal with mean zero and variance
1.
(Hint: by using the CLT, first show Zn is the sum
of a random sample of size n from a Poisson random variable with
mean 1.)

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 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.

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.

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