32. It may be shown that if Tn is any sequence of random variables such that TnT and if the variances of T and Tn exist, then lim inf, Var(T) ≥ Var(T). Let Tn = X1[|X| ≤ 1 − n¯¹] + n1[|X| > 1 − n¯¹] where X is uniform, U(-1, 1). Show that as n → ∞, Tn ✩ X, but Var(T) → ∞.
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![32. It may be shown that if Tn is any sequence of random variables such that TT and
if the variances of T and T exist, then lim infn Var(Tn) ≥ Var(T). Let
Tn = X1[|X| ≤ 1 − n¯¹] +n1[|X| > 1 − n¯¹]
where X is uniform, U(−1,1). Show that as n → ∞, Tn ✩ X, but Var(T) → ∞.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66f52ea3-13fd-4902-bc86-2f48a4cfbf0d%2F5d7cb6ba-54f5-45d0-abc9-586c8d0200e9%2F2qg6sbv_processed.png&w=3840&q=75)
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- Let X1, X2,..., Xn be a random sample from a population with a pdf a+2022 (a+ 2021) („) ;x > 0, a > 0 x+1 f (x) 0, elsewhere. II n 1 Show that | is sufficient for a. X; +1Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]Let f(x, y) = x + y for 0 < x < 1 and 0 < y < 1 The Conditional Variance of Y when X = ; is
- Let X1 be the sample mcan of a random sample of size n from a Norinal(jµ, o†) population and X2 be the sample mean of a random sample of the same size n from a Normal(j1, o,) population and the two samples are independent. Note that the two samples have the same population Imean ji but o + o3. Let i = wX1 +(1 – w)X2,0Assume x-Exp(1) and Y~N(0,x), that is Y is normally distributed with a random variance. Use Monte Carlo estimation to estimate [R Programming]State the CLT for independent identically distributed real RVs with mean u and variance a². Suppose that X. X,. . . are independent identically distributed random variables each uniformly distributed over the interval [0, 1]. Calculate the mean and variance of In X. Suppose that 0Suppose Y is a continuous random variable drawn from the uniform distributionon the interval [3, 4], that is, Y ∼ Uniform([3, 4]). Conditioned on Y = y, a second randomvariable X is drawn from the uniform distribution on the interval [0, y]. What is fX(x), thepdf of X?b) Let X,, X2,...,X, be a random sample from b(1, p). It is known that X is an p(1-p) unbiased estimator of p and that Var(X) = i. Find the Cramer-Rao lower bound for the variance of every unbiased estimator of p given that I(0) p(1-p) ii. What is the efficiency of X as an estimator of p?Let X;, i = 1,2, .. be independent and identically distributed Bernoulli(p) random variables. Define a) Show that yn(X – p) → N(0, p(1 – p)) in distribution. | The variance of a Bernoulli(p) is p(1– p). We might estimate this using X(1-X). For p + 1/2, find the asymptotic distribution of X(1 – X). Hint: make sure you use the Delta method! b)Let X1, X2, ..., X, be a random sample from a population r.v. X, with range Rx c [a, b]. Furthermore, suppose the mean and variance of X are unknown. Assume that n is large enough. (b – a)? (a) Show that Var(X) < b – a (b) Let a € (0, 1). Show that (X- 24 2n b a where (ze) = 1- is a confidence interval for E[X] with (1 – a)100% confidence level.Let X be a continuous random variable with the followirg function. Find: (a) P(2 6). * f(x) = 0. (1/8 if 0 < x <8 elsewhere6. Let X1,..., Xn ~ ƒ be an iid random sample with mean µ and variance o². (a) Find E(X; – X;)² for i + j. -SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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