3. Let Yx Poisson(j). That is, PY - Y|H) – 4P . Let, Gamma(0,B), i.e., f04)...
3. Let Yx Poisson(j). That is, PY - Y|H) – 4P . Let, Gamma(0,B), i.e., f(x) - "-exp(-B1). Find the marginal distribution of Y, i.e., find P(Y - y)
Let y,p ~iid Exp (0), for i = 1, . . . , n. (p(y|0) for 6 to be Gamma(a, b), tha distribution of θ BeAy). Assume the prior distribution Find the posterior 2. t is, p(0) -ba/ra)ge-i exp{-be. 3. Find the posterior predictive distribution of a future observation in problem 2
iid 14 marksAssume that e Denote T 4i Gamma(k, A) and X1,... , X,,e Poisson(0) (a) [4 marks Show that the posterior distribution of 0 is Gamma(nTk, n ). (b) [4 marks Find the probability function of the marginal distribution of Y = nX. (Note that the conditional distribution of on Y is not the same X1, ..., Xn.) as on iid 14 marksAssume that e Denote T 4i Gamma(k, A) and X1,... , X,,e Poisson(0) (a) [4 marks Show...
Let X, Y and Z be three independent Poisson random variable with parameters λι, λ2, and λ3, respectively. For y 0,1,2,t, calculate P(Y yX+Y+Z-t) (Hint: Determine first the probability distribution of T -X +Y + Z using the moment generating function method. Moment generating function for Poisson random variable is given in earlier lecture notes) Let X, Y and Z be three independent Poisson random variable with parameters λι, λ2, and λ3, respectively. For y 0,1,2,t, calculate P(Y yX+Y+Z-t) (Hint:...
(20 points) Consider the following joint distribution of X and Y ㄨㄧㄚ 0 0.1 0.2 1 0.3 0.4 (a) Find the marginal distributions of X and Y. (i.e., Px(x) and Py()) (b) Find the conditional distribution of X given Y-0. (i.e., Pxjy (xY-0)) (c) Compute EXIY-01 and Var(X)Y = 0). (d) Find the covariance between X and Y. (i.e., Cov(X, Y)) (e) Are X and Y independent? Justify your answer. (20 points) Consider the following joint distribution of X and...
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,...
(a)Suppose X ∼ Poisson(λ) and Y ∼ Poisson(γ) are independent, prove that X + Y ∼ Poisson(λ + γ). (b)Let X1, . . . , Xn be an iid random sample from Poisson(λ), provide a sufficient statistic for λ and justify your answer. (c)Under the setting of part (b), show λb = 1 n Pn i=1 Xi is consistent estimator of λ. (d)Use the Central Limit Theorem to find an asymptotic normal distribution for λb defined in part (c), justify...
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 >...
3. Let G1 ∼ Gamma(α1, β) and G2 ∼ Gamma(α2, β) and let G1 and G2 be independent. Define B1 = G1/(G1 + G2) and B2 = G1 + G2. (a) Find the joint pdf of (B1, B2). (b) Give the marginal pdf of B1 and identify its distribution. (c) Give the marginal pdf of B2 and identify its distribution.
As described in class, the Poisson Bracket [F, G] between two functions Fand G of the generalized positions q, and momenta pi is defined as: Consider a system with Hamiltonian H-P2/2m-Vr = (P, 2+py 2+pz2y2m)-y(x"2 + y"2 + z ^2)-U2 where yis a constant. a) Evaluate [Lz, H] and interpret the result in two ways i.e. what it says about L, and what it says about H b) Using the Poisson Bracket and the given Hamiltonian, find the value of...