Homework Answers
TOPIC:MLE, Normal distribution and Confidence interval.
Add Answer to:
Let X1 and X2 be independent random variables so X1~ N(u,1) and X2 N(u,4) Where u...
Let X1, X2, · · · be independent random variables, Xn ∼ U(−1/n, 1/n). Let X...
Let X1, X2, · · · be independent random variables, Xn ∼ U(−1/n, 1/n). Let X be a random variable with P(X = 0) = 1. (a) what is the CDF of Xn? (b) Does Xn converge to X in distribution? in probability?
Let Xi, X2,... , Xn denote independent and identically distributed uniform random variables on the interval...
Let Xi, X2,... , Xn denote independent and identically distributed uniform random variables on the interval 10, 3β) . Obtain the maxium likelihood estimator for B, B. Use this estimator to provide an estimate of Var[X] when r1-1.3, x2- 3.9, r3-2.2
5. Let X; (i = 1, 2, 3) be be independent gamma random variables with a;...
5. Let X; (i = 1, 2, 3) be be independent gamma random variables with a; = i and B. = 8. a. Find a maximum likelihood estimator of 8 and prove that it is unbiased. b. Show that 2(X1+X2+Xa) is a pivotal quantity for 0. c. Find a 95% confidence interval for 6.
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a)...
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...
Let X1 and X2 be independent n(0,1) random variables. Find the pdf of (X1 - X2)^2/2
Let X1 and X2 be independent n(0,1) random variables. Find the pdf of (X1 - X2)^2/2
Let X1 and X2 be random variables, not necessarily independent. Show that E [X1 + X2]...
Let X1 and X2 be random variables, not necessarily independent. Show that E [X1 + X2] = E [X1] + E [X2]. You may assume that X1 and X2 are discrete with a joint probability mass function for this problem, while the above inequality is true also for continuous random variables.
Suppose that X1,X2,. X are iid random variables with pdf ,220 (a) Find the maximum likelihood...
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).
8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that...
8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that M. 1, that is, M, converges in probability to 1 as n o . - Show that n(1 - M.) Exp(1), that is, n(1 - M.) converges in distribution to an exponential r.v. with mean 1 as n .
(10 marks) Let X1, X2,... be a sequence of independent and identically distributed random variables with...
(10 marks) Let X1, X2,... be a sequence of independent and identically distributed random variables with mean EX1 = i and VarX1 = a2. Let Yı, Y2, ... be another sequence of independent and identically distributed random variables with mean EY = u and VarY1 a2 Define the random variable ( ΣxΣ) 1 Dn 2ng2 i= i=1 Prove that Dn converges in distribution to a standard normal distribution, i.e., prove that 1 P(Dn ) dt 2T as n >oo for...
4 points) Let X1, X2 be independent random variables, with X1 uniform on (3,9) and X2...
4 points) Let X1, X2 be independent random variables, with X1 uniform on (3,9) and X2 uniform on (3, 12). Find the joint density of Y = X/X2 and Z = Xi X2 on the support of Y, Z. f(y, z) =
Let Xi, X2,... , Xn denote independent and identically distributed uniform random variables on the interval 10, 3β) . Obtain the maxium likelihood estimator for B, B. Use this estimator to provide an estimate of Var[X] when r1-1.3, x2- 3.9, r3-2.2
5. Let X; (i = 1, 2, 3) be be independent gamma random variables with a; = i and B. = 8. a. Find a maximum likelihood estimator of 8 and prove that it is unbiased. b. Show that 2(X1+X2+Xa) is a pivotal quantity for 0. c. Find a 95% confidence interval for 6.
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...
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).
8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that M. 1, that is, M, converges in probability to 1 as n o . - Show that n(1 - M.) Exp(1), that is, n(1 - M.) converges in distribution to an exponential r.v. with mean 1 as n .
(10 marks) Let X1, X2,... be a sequence of independent and identically distributed random variables with mean EX1 = i and VarX1 = a2. Let Yı, Y2, ... be another sequence of independent and identically distributed random variables with mean EY = u and VarY1 a2 Define the random variable ( ΣxΣ) 1 Dn 2ng2 i= i=1 Prove that Dn converges in distribution to a standard normal distribution, i.e., prove that 1 P(Dn ) dt 2T as n >oo for...
4 points) Let X1, X2 be independent random variables, with X1 uniform on (3,9) and X2 uniform on (3, 12). Find the joint density of Y = X/X2 and Z = Xi X2 on the support of Y, Z. f(y, z) =
Most questions answered within 3 hours.
-
Explain in detail
Germany is the fifth largest economy
explain what goods and services Germany specializes...
asked 3 minutes ago -
The density of platinum is 21.45 g/mL. If a cube of platinum
with a mass of...
asked 8 minutes ago -
Accounts Receivable
Sales
A/R Posting
Extended Sales Invoice
Packing Slip
Compare invoice to packing slip 2...
asked 11 minutes ago -
Michaella, age 23, is a full-time law student and is claimed by
her parents as a...
asked 12 minutes ago -
Why are polymers not typically casted into products?
asked 29 minutes ago -
When rolling a die 129 times, what is the probability of rolling
a 6 no more...
asked 46 minutes ago -
4. A call option currently sells for $7.75. It has a strike
price of $85 and...
asked 35 minutes ago -
1.
You need to prepare 10.0 liters of an acid aqueous solution with a
pH of...
asked 37 minutes ago -
Along an aggregate supply curve, if the level of output is less
than the natural level...
asked 38 minutes ago -
By 2025, annual consumption in emerging markets will total $30
trillion and contribute more than ________...
asked 43 minutes ago -
At what point does reformation cease to be a viable option for
those who are oppressed...
asked 47 minutes ago -
Place letters corresponding to amounts in the proper order for
lightest to heaviest samples:
a) 2100...
asked 51 minutes ago