Question

Data yi, i = 1, . . . , n arise from a Poisson distribution with rate parameter λ.

Show that the posterior distribution for λ|y1,...,yn is also Gamma distributed when a Gamma(α,β) prior is used.

Answer #1

Data yi, i = 1, . . . , n arise from a Poisson distribution with
rate parameter λ.
If the data are
y = 17,25,25,21,13,22,23
find the posterior for λ given the above specified Gamma prior.
Comment on the posterior, data, and prior means.

Data yi, i = 1, . . . , n arise from a Poisson
distribution with rate parameter λ.
(a) Write down the likelihood for the model, up to the constant
of proportionality.
(b) A Gamma distribution is proposed as the prior. How can we
use such a prior to include our belief that λ is 10±1 i.e. mean 10
and standard deviation 1?

Data yi, i = 1, . . . , n arise from a Poisson
distribution with rate parameter λ.
(a) Write down the likelihood for the model, up to the constant
of proportionality.
(b) A Gamma distribution is proposed as the prior. How can we
use such a prior to include our belief that λ is 10±1 i.e. mean 10
and standard deviation 1?

Let X1, . . . , Xn be iid from a Poisson distribution with
unknown λ. Following the Bayesian paradigm, suppose we assume the
prior distribution for λ is Gamma(α, β).
(a) Find the posterior distribution of λ.
(b) Is Gamma a conjugate prior? Explain.
(c) Use software or tables to provide a 95% credible interval
for λ using the 2.5th percentile and 97.5th percentile in the case
where xi = 13 and n=10, assuming α = 1 andβ =...

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 .

The special case of the gamma distribution in which α is a
positive integer n is called an Erlang distribution. If we replace
β by 1 λ in the expression below, f(x; α, β) = 1 βαΓ(α) xα − 1e−x/β
x ≥ 0 0 otherwise the Erlang pdf is as follows. f(x; λ, n) = λ(λx)n
− 1e−λx (n − 1)! x ≥ 0 0 x < 0 It can be shown that if the times
between successive events are...

suppose we draw a random sample of size n from a Poisson
distribution with parameter λ. show that the maximum likelihood
estimator for λ is an efficient estimator

We write ? ∼ Poisson (?) if ? has the Poisson distribution with
rate ? > 0, that is, its p.m.f. is ?(?|?) = Poisson(?|?) = ?
^??^x /?!
Assume a gamma distribution as the prior for ? where ?(?) = ?
^??(?) ? ^?-1e ^??
?> 0 Use Bayes Rule to derive the posterior distribution
?(?|?).
b. Let’s reconsider the car accidents example introduced in
classed. Suppose that (X) the number of car accidents at a fixed
point on...

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, 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...

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