Let
be i.i.d.
. Define the sample mean and the sample variance by
and
.
(i) Find the distribution of and
for i = 1, ... , n.
(ii) Show that and
are independent for i = 1, ... , n.
(iii) Hence, or otherwise, show that and
are
independent.
Let be i.i.d. . Define the sample mean and the sample variance by and . (i)...
.........,,
random sampling of the normal distribution of the unit n, and and
ile
-1 Let the sample mean and sample variance be
respectively.
a)
b)
ile
-1 it is independent.
c)
d)
What is the proof ??
We were unable to transcribe this imagen Ν(μ, σ2) T We were unable to transcribe this imageWe were unable to transcribe this imageT We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable...
Exercise 5 (Sample variance is unbiased). Let X1, ... , Xn be i.i.d. samples from some distribution with mean u and finite variance. Define the sample variance S2 = (n-1)-1 _, (Xi - X)2. We will show that S2 is an unbiased estimator of the population variance Var(X1). (i) Show that ) = 0. (ii) Show that [ŠX – 1908–) -0. ElCX –po*=E-* (Šx--) == "Varex). x:== X-X+08 – ) Lx - X +2Zx - XXX - 1) + X...
3. Let ,..., be
independent random sample from N(),
where is unknown.
(i) Find a sufficient statistic of .
(ii) Find the MLE of .
(iii) Find a pivotal quantity and use it to construct a
100(1–)% confidence
interval for .
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable...
Let X1, X2,.......Xn be a
random sample of size n from a continuous distribution symmetric
about .
For testing H0: =
10 vs H1: <
10, consider the statistic T- =
Ri+ (1-i),
where i
=1 if Xi>10 , 0 otherwise; and
Ri+ is the rank of (Xi - 10) among
|X1 -10|, |X2-10|......|Xn
-10|.
1. Find the null mean and variance of T- .
2. Find the exact null distribution of T- for
n=5.
We were unable to transcribe this imageWe were...
Let be iid observations from , is known and is an unknown real number. Let be the parameter of interest. (a) Find the CRLB for the variance of an unbiased estimator for . (b) Find the UMVUE for . (c) Show that is an unbiased estimator for . (d) Show that . We were unable to transcribe this imageσ2 (μ, ) We were unable to transcribe this imageWe were unable to transcribe this imageg(t) = 211 We were unable to...
Let X = (X1, . . . , Xn) be a random sample of size n with mean μ and variance σ2. Consider Tm i=1 (a) Find the bias of μη(X) for μ. Also find the bias of S2 and ỡXX) for σ2. (b) Show that Hm(X) is consistent. (c) Suppose EIXI < oo. Show that S2 and ỡXX) are consistent.
Let X = (X1, . . . , Xn) be a random sample of size n with mean μ...
Let X1,...,X10 be a random sample from N(θ1,1) distribution and let Y1,...,Y10 be an independent random sample from N(θ2,1) distribution. Let φ(X,Y ) = 1 if X < Y , −5 if X ≥ Y , and V= φ(Xi,Yj) . 1. Find v so that P[V>=v]=0.45 when 1=2. 2. Find the mean and variance of V when 1=2. 10 10 2 We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe...
Question 4. Suppose for i=1,...,n both the mean and variance are unknown. Based on n=100 sample data, we would like to test vs a) at a type 1 error level , find a sample statistic T and the rejection region R that correctly controls exactly, i.e., find T and R that satisfy (must be exact in distribution not approximate). b) Compute the asymptotic power of T, i.e., what does converge to as sample size goes to infinity? Question 5. Following...
Let , ... be independent random variables with mean zero and finite variance. Show that We were unable to transcribe this imageWe were unable to transcribe this image
2. (10pts) Let X1, X2, , X20 be an i.i.d. sannple from a Normal distribution with mean μ and variance σ2, ie., Xi, X2, . . . , X20 ~ N(μ, σ2), with the density function Also let 20 20 10 20 -20 19 i-1 ー1 (a) (5pts) What are the distributions of Xi - X2 and (X1 - X2)2 respectively? Why? (b) (5pts) what are the distributions of Y20( and 201 ? Why? (X-μ)2
2. (10pts) Let X1, X2,...