Homework Answers
10.
Here 
And T can expressed as 
Z~ N(0,1) and W~
clearly we know that If Z and W are independently distributed then the variable T follows a t distribution with degrees of freedom (n-1)
~ 
means T has a t-distribution with n - 1 degrees of
freedom
So, our answer is option (C)
Answer: (C) both (A) and (B)
Add Answer to:
v. suppose that X1,...,x, is a random sample with a common Nu d istribution. sample mean...
and let (b) Let X, X,...,X, be a random sample form the normal distribution Nu,o) Σ-...
and let (b) Let X, X,...,X, be a random sample form the normal distribution Nu,o) Σ- ΣΧ be the sample mean, S2 be the sample variance. j-1 n-1 Σ--Σ( - 1' -nΣΤ-β). (i) Prove that Using it, determine the distribution of X (ii) Find the m.g.f. of X. n ΣT- ) Σ- 7 7 n (iii) Indicate the distributions ofJ 2 , respectively. and (iii) Given that X and S are independent, derive the m.g.f of (n-15, and then, σ'...
Let X1, X2,..., X, be n independent random variables sharing the same probability distribution with mean...
Let X1, X2,..., X, be n independent random variables sharing the same probability distribution with mean y and variance o? (> 1). Then, as n tends to infinity the distribution of the following random variable X1 + X2 + ... + x, nu vno converges to Select one: A. an exponential distribution B. a normal distribution with parameters hi and o? C a normal distribution with parameters 0 and 1 D. a Poisson distribution
Proof this Theorem 2.1 Let X1,...,X, be a random sample from a population with mean J...
Proof this
Theorem 2.1 Let X1,...,X, be a random sample from a population with mean J and variance o? < 2. Then (i) E(7) = j, var(7) = ?, and E(S2) = 02. (ii) The moment generating function (m.g.f.) of X is Mg(t) = [My(t/n)]”, where My(t) = E(e*X) is the m.g.f. of X.
A random sample of 49 measurements from one population had a sample mean of 13, with...
A random sample of 49 measurements from one population had a sample mean of 13, with sample standard deviation 3. An independent random sample of 64 measurements from a second population had a sample mean of 15, with sample standard deviation 4. Test the claim that the population means are different. Use the level of significance 0.01. Using s1 = 3 and s2 = 4, we can compute the t value corresponding to the test statistic x1 − x2 = −2. Recall...
Let X = (X1, . . . , Xn) be a random sample of size n with mean μ and variance σ2. Consider Tm i=...
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...
Please show step by step solution. 7. Let X1, X2, ..., Xn be i.i.d. random variables...
Please show step by step solution.
7. Let X1, X2, ..., Xn be i.i.d. random variables drawn from a N(u,0%). Show that the Sample Variance (52) and the Maximum Likelihood Estimator (S) of o2 are both Consistent Estimators for o?. S2 27=2(X-X)2 and S 21-2(X;-) n-1 n (n-1)S Hint: has a Chi-Square Distr. with (n − 1) degrees of freedom. E(x{n-1)) = n-1,V(xin-1)) = 2(n − 1)
Suppose X1,X2, .. ,X, is a random sample from a standard normal distribution and let Z...
Suppose X1,X2, .. ,X, is a random sample from a standard normal distribution and let Z be another standard normal variable that is independent of X1, X2, .., X,. 9 9 9 Determine the distribution of each of the variables X, U and V. (a) (b) Determine the distribution of the variable 3Z NU Determine the distribution of the variable W- (c) (d) Determine the distribution of the variable R -4y (where Y is the variable from (C)
Please answer question (a) X1 - X X2 – Å a. Let X1, ..., Xn i.i.d....
Please answer question (a)
X1 - X X2 – Å a. Let X1, ..., Xn i.i.d. random variables with X; ~ N(u, o). Express the vector in the | Xn – form AX and find its mean and variance covariance matrix. Show some typical elements of the vari- ance covariance matrix. b. Refer to question (a). The sample variance is given by S2 = n11 21–1(X; – X)2, which can be ex- pressed as S2 = n1X'(I – 111')X (why?)....
Suppose that X1,X ..Xn is a set of independent random variables each arising from the same Bernou...
Suppose that X1,X ..Xn is a set of independent random variables each arising from the same Bernoulli distribution with mean p. The estimator X, where X is the sample mean, has mean p an variance p(1-p)/n. Consider an estimator of p of the form where K is a constant. This estimator has mean p+K and variance p(1 -p)/n. Select the option that gives the mean squared error of WK
Suppose that X1,X ..Xn is a set of independent random variables...
Assume X1, . . . , Xn iid normal with mean and variance ^2 , show...
Assume X1, . . . , Xn iid normal with mean
and variance
^2 , show that
a. X¯ and X^2 are independent.
b. Proof that X¯ is normally distributed with mean
and variance
^2/n.
c. Proof that (n ? 1)S^2/?2 is chi-squared distributed with (n ?
1) degrees of freedom.
d. Show that X¯ S/is
t distributed with (n ? 1) degrees of freedom
and let (b) Let X, X,...,X, be a random sample form the normal distribution Nu,o) Σ- ΣΧ be the sample mean, S2 be the sample variance. j-1 n-1 Σ--Σ( - 1' -nΣΤ-β). (i) Prove that Using it, determine the distribution of X (ii) Find the m.g.f. of X. n ΣT- ) Σ- 7 7 n (iii) Indicate the distributions ofJ 2 , respectively. and (iii) Given that X and S are independent, derive the m.g.f of (n-15, and then, σ'...
Let X1, X2,..., X, be n independent random variables sharing the same probability distribution with mean y and variance o? (> 1). Then, as n tends to infinity the distribution of the following random variable X1 + X2 + ... + x, nu vno converges to Select one: A. an exponential distribution B. a normal distribution with parameters hi and o? C a normal distribution with parameters 0 and 1 D. a Poisson distribution
Proof this
Theorem 2.1 Let X1,...,X, be a random sample from a population with mean J and variance o? < 2. Then (i) E(7) = j, var(7) = ?, and E(S2) = 02. (ii) The moment generating function (m.g.f.) of X is Mg(t) = [My(t/n)]”, where My(t) = E(e*X) is the m.g.f. of X.
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...
Please show step by step solution.
7. Let X1, X2, ..., Xn be i.i.d. random variables drawn from a N(u,0%). Show that the Sample Variance (52) and the Maximum Likelihood Estimator (S) of o2 are both Consistent Estimators for o?. S2 27=2(X-X)2 and S 21-2(X;-) n-1 n (n-1)S Hint: has a Chi-Square Distr. with (n − 1) degrees of freedom. E(x{n-1)) = n-1,V(xin-1)) = 2(n − 1)
Suppose X1,X2, .. ,X, is a random sample from a standard normal distribution and let Z be another standard normal variable that is independent of X1, X2, .., X,. 9 9 9 Determine the distribution of each of the variables X, U and V. (a) (b) Determine the distribution of the variable 3Z NU Determine the distribution of the variable W- (c) (d) Determine the distribution of the variable R -4y (where Y is the variable from (C)
Please answer question (a)
X1 - X X2 – Å a. Let X1, ..., Xn i.i.d. random variables with X; ~ N(u, o). Express the vector in the | Xn – form AX and find its mean and variance covariance matrix. Show some typical elements of the vari- ance covariance matrix. b. Refer to question (a). The sample variance is given by S2 = n11 21–1(X; – X)2, which can be ex- pressed as S2 = n1X'(I – 111')X (why?)....
Suppose that X1,X ..Xn is a set of independent random variables each arising from the same Bernoulli distribution with mean p. The estimator X, where X is the sample mean, has mean p an variance p(1-p)/n. Consider an estimator of p of the form where K is a constant. This estimator has mean p+K and variance p(1 -p)/n. Select the option that gives the mean squared error of WK
Suppose that X1,X ..Xn is a set of independent random variables...
Assume X1, . . . , Xn iid normal with mean
and variance
^2 , show that
a. X¯ and X^2 are independent.
b. Proof that X¯ is normally distributed with mean
and variance
^2/n.
c. Proof that (n ? 1)S^2/?2 is chi-squared distributed with (n ?
1) degrees of freedom.
d. Show that X¯ S/is
t distributed with (n ? 1) degrees of freedom
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