Question

Let Y ⇠ Gamma(alpha,beta) and conditioned on Y = y, X
⇠ Poisson(y).

Find the unconditional distribution of X in the case that alpha = r
is an integer and beta=1-p/p
for p in (0, 1).
Find the conditional
distribution of Y|X = x. (Use Bayes’ rule)

Answer #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...

For the hierarchical model Y |Λ ∼ Poisson(Λ) and Λ ∼ Gamma(α,
β), find the marginal distribution, mean, and variance of Y . Show
that the marginal distribution of Y is a negative binomial if α is
an integer. (b) Show that the three-stage model Y|N∼Binomial(N,p),
N|Λ∼Poisson(Λ), andΛ∼Gamma(α,β) leads to the same marginal
distribution of Y .

Let N have a Poisson distribution with parameter lander=1.
Conditioned on N=n,
let X have a uniform distribution over the integers
0,1,.......,n+1. What is the
marginal distribution for X?
Step by step and show what definition you use

Let X ~ beta(alpha, beta). Find the population mode of X for
alpha > 1 and beta > 1.

Let X be a gamma random variable with parameters alpha = 4 and
beta = 4. Using Markov's inequality, calculate an upper bound for
the probability that X is greater than or equal to 10.

Consider Poisson distribution f(x|θ) = (e^−θ) [(θ^x) / (x!)] for
x = 0, 1, 2, . . .
Let the prior distribution for θ be f(θ) = e^−θ for θ >
0.
(a) Show that the posterior distribution is a Gamma
distribution. With what parameters?
(b) Find the Bayes’ estimator for θ.

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).

Let X1, X2 be a sample of size 2 from the Gamma (Alpha=2, Lamba
= 1/theta) distribution
X1 = Gamma = x/(theta^2) e^(-x/theta)
Derive the joint pdf of Y1=X1 and Y2 = X1+X2
Derive the conditional pdf of Y1 given Y2=y2. Can you name that
conditional distribution? It might not have name

Let X,..., Xn be exponential with mean beta. Find
UMVUEs for beta, beta^2, beta^3. (Use the version of the
exponential distribution with PDF p(x)= 1/beta e^(-x/beta)
(x>0), and so Mx(t)=(1-beta(t))^-1.)

3. (10pts) Let Y be a continuous random variable having a gamma
probability distribution with expected value 3/2 and variance 3/4.
If you run an experiment that generates one-hundred values of Y ,
how many of these values would you expect to find in the interval
[1, 5/2]?
4. (10pts) Let Y be a continuous random variable with density
function f(y) = 1 2 e −|y| , −∞ < y < ∞ 0, elsewhere (a) Find
the moment-generating function of...

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