3. Suppose that X1,X2, ,Xn are i.id. N(0, σ2). Find a function of T(X)-Σǐii verges in...
tirling's Formula, which gives approximation for factorials, can be derived using CLT. (a) Suppose that X1, X2, random variable Z, , Xn is an ii.d. sample from Exp(1). Show that, for a standard normal (b) Show by differencing both sides of the approximation in part a. Then set 0 to get Stirling's Formula.
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
(b) For n = 100, give an approximaation for P(Y> 100) (c) Let X be the sample mean, then approximate P(1.1< 1.2) for -100. 2. Consider a random sample XX from CDF F(a) 1-1/ for z [1, 0o) and zero otherwise. (a) Find the limiting distribution of XiI.n, the smallest order statistic. (b) Find the limiting distribution of XI (c) Find the limiting distribution of n In X1:m- 3. Suppose that X,,, are iid. N(0,o2). Find a function of T(x)x...
(a) Suppose that i, X2,... , In is an i.i.d. sample from Exp(1). Show that, for a standard normal random variable Z b) Show Г(n) by differencing both sides of the approximation in part a. Then set a -0 to get Stirling's Formula. 5. Suppose that Y is an id sample from Negative Binomial (n,p). Give a normal approximation of Yn use CLT, when n is large. 6. (Mandatory for Graduate Student. Extra credit for undergrad.) Let Ai, converges to...
Stirling’s Formula, which gives approximation for factorials, can be derived using CLT. (a) Suppose that X1, X2, · · · , Xn is an i.i.d. sample from Exp(1). Show that, for a standard normal random variable Z, (b) Show by differencing both sides of the approximation in part a. Then set x = 0 to get Stirling’s Formula. We were unable to transcribe this imageГ(n) уж
Let X1, X2, . . . , Xn be IID N(0, σ2 ) variables. Find the rejection region for the likelihood ratio test at level α = 0.1 for testing H0 : σ2 = 1 vs H1 : σ2 = 2.
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...
8. Let X, X2, , xn all be be distributed Normal(μ, σ2). Let X1, X2, , xn be mu- tually independent. a) Find the distribution of U-Σǐ! Xi for positive integer m < n b) Find the distribution of Z2 where Z = M Hint: Can the solution from problem #2 be applied here for specific values of a and b?
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
7.6.4. Let X1, X2,... , Xn be a random sample from a uniform (0,) distribution. Continuing with Example 7.6.2, find the MVUEs for the following functions of (a) g(0)-?2, i.e., the variance of the distribution (b) g(0)- , i.e., the pdf of the distribution C) or t real, g(9)- , î.?., the mgf of the distribution. Example 7.6.2. Suppose X1, X2,... , Xn are iid random variables with the com- mon uniform (0,0) distribution. Let Yn - max{X1, X2,... ,...