The Poisson distribution is a useful discrete distribution which can be used to model the number ...
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume the prior distribution of Of λ is is an exponential with mean 1, i.e. the prior pdi g(A) e-λ, λ > 0 Note that the exponential distribution is a special gamma distribution; and a general gamma distribution with parameters α > 0 and β > 0 has the pd.f. h(A; α, β)-16(. otherwise Also the mean of a gamma random variable with the pd.f.h(Χα,...
Please answer the question clearly. Consider a random sample of size n from a Poisson population with parameter λ (a) Find the method of moments estimator for λ. (b) Find the maximum likelihood estimator for λ. Suppose X has a Poisson distribution and the prior distribution for its parameter A is a gamma distribution with parameters and β. (a) Show that the posterior distribution of A given X-x is a gamma distribution with parameters a +r and (b) Find the...
11. Obtain the MLE estimate for the beta parameter in Gamma distribution defined below for n iid (identical and independent) observations in a sample. Show steps. Obtain the MLE estimate for the alpha parameter. The continuous random variable X has a gamma distribution, with param eters α and β, if its density function is given by x>0, elsewhere, .tor"-le-z/ß, f(x; α, β)-Ί where α > 0 and β > 0. (You will also need the beta estimate, use the direct...
This question uses a discrete probability distribution known as the Poisson distribution. A discrete random variable X follows a Poisson distribution with parameter λ if Pr(X = k) = Ake-A ke(0, 1,2, ) k! You are a warrior in Peter Jackson's The Hobbit: Battle of the Five Armies. Because Peter decided to make his battle scenes as legendary as possible, he's decided that the number of orcs that will die with one swing of your sword is Poisson distributed (lid)...
1. Suppose that X P(A), the Poisson distribution with mean λ Assuming squared error loss, derive that Bayes estimator of λ with respect to the prior distribution「(Q), first by explicitly deriving the marginal probability mass function of X, obtaining an expression for the posterior density of A and evaluating E(alx) and secondly by identifying g(Alx) by inspection and noting that it is a familiar distribution with a known mean.
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each other. Given an outcome x-(xi, , X.), the objective is to estimate both λ and p. Consider the following hierarchical Bayesian model: P U(0, 1) alp) Gammala, b) rlp.i)~Ber(p independently (x,lr.λ.Ρ) ~ Poiar.) independently . where r () and a and b are known parameters. We...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each other. Given an outcome x-(xi, , X.), the objective is to estimate both λ and p. Consider the following hierarchical Bayesian model: P U(0, 1) alp) Gammala, b) rlp.i)~Ber(p independently (x,lr.λ.Ρ) ~ Poiar.) independently . where r () and a and b are known parameters. We...
have a Ba- Question 2. (20 points) Let (X1,...,xn) be a sample from Poisson(2), and let prior distribution Gamma(a,b), a,ß > 0, with pdf fe exp(-82), when 1 2 0, (A) =1 Ta) * 10, when 1 <0, where I(a):= 6°40-1 exp(-t)dt, for a > 0. (a). (10 points) Find the posterior distribution of 1. (0.1) (b). (5 points) Calculate the posterior mean and variance. (c). (5 points) Now consider a Bayesian test of H: 15 lo versus H:2 >...
Number 4 turns out to be an inverse gamma function with parameters alpha= n and beta= the sum of x sub i PLEASE ANSWER #5 NOT #4 4. Suppose that X1,X2, 10 pts. the p.d.f. is given by form a random sample from a distribution for which where the unknown parameter θ > 0. Suppose also that the improper prior of θ is m(0) Find the posterior distribution π(θ x). Hint: The inverse gamina distribution from question 6 in Homework...
In each of the summer months (June, July, August), the number of accidents per months at a busy intersection is Poisson distributed with mean 1.5 accidents/month. For all other months, the number of accidents is Poisson distributed with mean 0.5 accidents/month. 7. a) (3 pts) First, let Y an, YFeb, YMar,... be the number of accidents occurring in the months of January, February, March, etc. Define a variable A- the total number of accidents occurring in the second half of...