Suppose x1, x2…x25 ~ iid N(2,100). Find
a. Pr(x̅< -1.5)
b. Pr(s2 < 45.235)
c. Pr[x̅< 2 + 0.3422(s)]
Suppose x1, x2…x25 ~ iid N(2,100). Find a. Pr(x̅< -1.5) b. Pr(s2 < 45.235) c. Pr[x̅<...
5.13. Suppose X1, X2, , xn are iid N(μ, σ2), where-oo < μ < 00 and σ2 > 0. (a) Consider the statistic cS2, where c is a constant and S2 is the usual sample variance (denominator -n-1). Find the value of c that minimizes 2112 var(cS2 (b) Consider the normal subfamily where σ2-112, where μ > 0. Let S denote the sample standard deviation. Find a linear combination cl O2 , whose expectation is equal to μ. Find the...
Suppose that X1,X2, ,Xn are iid N(μ, σ2), where both parameters are unknown. Derive the likelihood ratio test (LRT) of Ho : σ2 < σ1 versus Ho : σ2 > σ.. (a) Argue that a LRT will reject Ho when w(x)S2 2 0 is large and find the critical value to confer a size α test. (b) Derive the power function of the LRT
For the given data, find ∑x, n, and x̅: x1 = 16, x2 = 21, x3 = 20, x4 = 17, x5 = 18, x6 = 17, x7 = 17, x8 = 11
Suppose that X1,X2,. X are iid random variables with pdf ,220 (a) Find the maximum likelihood estimate of the parameter a (b) Find the Fisher Information of X1,X2,.. ., Xn and use it to estimate a 95% confidence interval on the MLE of a (c) Explain how the central limit theorem relates to (b).
Suppose that (X1, X2,,,,Xn) are iid random variables. Find the maximum likelihood estimator of theta for the following distributions 1) Poi(theta) 2) N(Mu, theta) 3) Exp(theta)
2. Suppose that X1, X2, . . . , Xn are iid. N(0, σ) with density function f (xlo) Find the Fisher information I(o) a. b. Now, call: σ2 your parameter, with this new parametrization, f(x19)-E-e-28 Find the Fisher information 1(8) 1(ог). Is 1(σ*)-1 (σ)? c. Find o2MOM d. Find σ2MLE e. Find Elo-MLE]. Show that σ2MLEls unbiased f. Find Var[σ 2MLEİ. Does σ2MLE attain the CRLB?
Delta Theorem's application to MLES Suppose X1, X2, ... are iid Poi(). a. Find the MLE of h() = P(X = 0) b. Find its asymptotic distribution
Suppose X1 and X2 are iid Poisson(θ) random variables and let T = X1 + 2X2. (a) Find the conditional distribution of (X1,X2) given T = 7. (b) For θ = 1 and θ = 2, respectively, calculate all probabilities in the above conditional distribution and present the two conditional distributions numerically.
Suppose X1, X2, ..., Xn are independent and identically distributed (iid) with a Uniform -0,0 distri- bution for some unknown e > 0, i.e., the Xi's have pdf Suppose X1, X2,..., Xn are independent and identically distributed (iid f(3) = S 20, if –0 < x < 0; 20 0, otherwise. (a) (4 pts) Briefly explain why or why not this is an exponential family (b) (5 pts) Find one meaningful sufficient statistic for 0. (By "meaningful”, I mean it...
t (0, c(X1-X2)2) įs a Let X, and X2 be iid. N(0, (Au)100% confidence interval for σ- 1) σ2) variables) . Find a constant so tha t (0, c(X1-X2)2) įs a Let X, and X2 be iid. N(0, (Au)100% confidence interval for σ- 1) σ2) variables) . Find a constant so tha