Question

Let X follow Poisson distribution with λ = a and Y follow Poisson distribution with λ = b. X and Y are independent. Define a new random variable as Z=X+Y. Find P(Z=k).

Answer #1

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 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

a) What are the modal values of a Poisson distribution X ~
P(λ)?
b) Y ~ P(λ) is independent from X ~ P(λ) (this is, identically
distributed like X). What is the probability distribution of Z = X
+ Y?

Poisson Distribution: p(x,
λ) = λx exp(-λ)
/x! , x = 0, 1, 2, …..
Find the moment generating function Mx(t)
Find E(X) using the moment generating function
2. If X1 , X2 ,
X3 are independent and have means 4, 9, and
3, and variencesn3, 7, and 5. Given that Y =
2X1 - 3X2 +
4X3. find the
mean of Y
variance of Y.
3. A safety engineer claims that 2 in 12 automobile accidents
are due to driver fatigue. Using the formula for Binomial
Distribution find the...

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, Y be random variables. Also let X|Y = y ~ Poisson(y) and
Y ~ gamma(a,b) is the prior distribution for Y. a and b are also
known.
1. Find the posterior distribution of Y|X=x where X=(X1, X2, ...
, Xn) and x is an observed sample of size n from the distribution
of X.
2. Suppose the number of people who visit a nursing home on a
day is Poisson random variable and the parameter of the Poisson...

Suppose X and Y are independent Poisson random variables with
respective parameters λ = 1 and λ = 2. Find the conditional
distribution of X, given that X + Y = 5. What distribution is
this?

a. Suppose X and Y are independent Poisson
random variables, each with expected value 2. Define Z=X+Y. Find
P(Z?3).
b. Consider a Poisson random variable X with
parameter ?=5.3, and its probability mass function, pX(x). Where
does pX(x) have its peak 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.
Derive the joint probability distribution function for X and Y.
Make sure to explain your steps.

Let X~Poisson(4) random variable and Y an independent
Bin(10,1/2) random variable.
(a) Use Markov's inequality to find an upper bound for P(X+Y
> 15).
(b) Use Chebyshev's inequality to find an upper bound for P(X+Y
> 15)

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