Problem 2. Bayesian signal extraction Suppose that X ∼ N (µ, σ2) and U ∼ N (0, τ 2) are independent. Let S = X + U. (i) Find the pdf of (X, S) and then identify this probability distribution; (ii) Find E[X | S = s]; (iii) Find Var[X | S]
Problem 2. Bayesian signal extraction Suppose that X ∼ N (µ, σ2) and U ∼ N...
1. Let X be an iid sample of size n from a continuous distribution with mean /i, variance a2 and such that Xi e [0, 1] for all i e {1,...,n}. Let X = average. For a E (0,1), we wish to obtain a number q > 0 such that: (1/n) Xi be the sample Р(X € |и — 9. и + q) predict with probability approximately In other words, we wish to sample of size n, the average X...
Let the random variable X follow a normal distribution with µ = 22 and σ2 = 7. Find the probability that X is greater than 10 and less than 17.
Question 1 Suppose x, = 1,2,3 has independent N(u,, 1) distributions such that 2, = , = 3u. Let 1 (2g + З23)?; q2 — k(3ӕ, — 2з)?; q 10 91= -G1) Suppose N (u, V); — (х1 х, х3); и 3 (M g and V x (central chi-squared distribution) 2p1 = plz = 3u3 and qa4 ~ (i) Determine whether q, has a chi-squared distribution (ii) Determine the degrees of freedom k and the noncentrality parameter A of q3...
Let X1, . . . , Xn ∼ independent N(µ, τ ) for µ known. Find the Likelihood Ratio Test for H0 : τ ≥ τ_0 vs H1 : τ < τ_0. Use the fact that MLE of τ is (1/n)S^2.
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (i) Write down the conditional probability density function of y given μ (ii) Show that rw1p) amp(剖-rr)
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (i) Write down the conditional probability density function of y given μ (ii) Show that...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (i) Write down the conditional probability density function of y given μ (ii) Show that rw1p) amp(剖-rr)
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (i) Write down the conditional probability density function of y given μ (ii) Show that...
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and μ is an unknown parameter (iii) Suppose that we have a prior μ ~ N(a, b-1) where b > 0, Show that the prior distribution π(A) verifies r(11) x exp (iv) Show that the posterior π(μ|y) verifies (v) which distribution is π(μ|y)?
Bayesian updating Suppose that we have the model y|μ ~ N(μ, τ-1) where τ > 0 is known and...
Consider the simplified Bayesian model for normal data The joint posterior pdf is ful, σ21 x)a(σ2,-/2-1 expl_jy.tx, _aPI The marginal posterior pdfs of μ and σ 2 can be obtained by integrating out the other variable (8.30) y@1 x) α (σ2)-m;,-1/2 expl-- Σ.-tri-x)2 (8.31) d. Let q1 and q2 be they/2 and 1-y/2 quantiles of (8.31). Show that the 1-γ credible interval (gi,q2) is identical to the classic confidence interval (5.19) (with ar replaced by y). Hence, a (1-α) stochastic...
5. [20+5+5] In the regression modely, x,B+ s, pe,+u, ,where I ρ k l and , , let ε, follow an autoregressive (AR) process u' ~ID(Qơ:) , t-l, 2, ,n . <l and u, - Derive the variance-covariance matrix Σ of (q ,6, , , ε" )". From the expression of Σ, identify and interpret Var(.) , t-1, 2, , n . Find the CorrG.ε. and explain its behavior as "s" increases, (s>0). (ii) (iii)
5. [20+5+5] In the regression...
5. A random variable X ∼ N (µ, σ2 ) is Gaussian distributed with mean µ and variance σ 2 . Given that for any a, b ∈ R, we have that Y = aX + b is also Gaussian, find a, b such that Y ∼ N (0, 1) Please show your work. Thanks!