Question

I REALLY need numbers 2 and 3 and 5 by like tomorrow morning. I have no clue how to do these. I know the image quality is iffy but please help as best you can

Homework1 STA4322 Homework 1, Spring 2019 Please turn in your own work, though you may discuss the problems with classmates, the TA, the Professor, the internet, etc. The most important thing is that you understand the problems and how they are solved as they will prepare you for the exam. Please turn in your work in class on Thunsday January 24th. If you can't come to class that day, please drop your homework in my mailbox (Griffin Floyd 103) by the time class has begun (1) The number of persons coming through a blood bank until the first person with blood type A is found is a random variable Y with a geometric(p) distribution. If p denotes the probability that any one randomly selected person will possess type A blood, then E(Y)-1/p and VY) Find a function of Y that is an unbiased estimator of V(Y) (2) Let Y be a binomial random variable with parameters n and p. Remember that We know that Y/n is an unbiased estimator of p. Now we want to estimate the variance of Y with n(X1-2) (a) Find the expected value of this esti (b) Find an unbiased estimator that is a simple modification of the proposed estimator mator (3) Suppose that E(h-θ, E(q-9, V(h-of, and V(02) σ2. Assume that θ1 and θ2 are independent. Consider the following estimator: (a) Show that 6, is unbiased for θ (b) Find the value of a that minimizes the variance of 03 (c) Which estimator would you use? θ., 02, or 6, when using the value of a found in part (b) (4) Let Yİ,. . .,Y, be N(0, 1). Iet 0-F, and 02-7 (a) What are the possible values of the 8 (b) Find the bias and MSE of both the estimators. (c) Is one of the estimators better than the other (d) For what values of θ is 0, better than 02? (5) Let Y...Y be independent random variables from a distribution with distribution function P(Y Sy- F), and density function f(). Now let Yay be the minimum of all the observations. Show that the density function of Y is given by Hint: First write out the CDF, P(Y S), then using independence of the observations put it in terms of the distribution function Fr"丶 und ,hamtake the derivative to get the density I 1 of 1

Answer 1

(2)

(a)

Estimator is,

$\hat{V} = n\left ( \frac{Y}{n} \right )\left (1- \frac{Y}{n} \right ) = n \left (\frac{Y}{n}- \frac{Y^2}{n^2} \right )$

Expected value of the estimator is,

$E[\hat{V} ]= E\left [ n \left (\frac{Y}{n}- \frac{Y^2}{n^2} \right ) \right ] = nE\left [ \frac{Y}{n} \right ] - n E\left [ \frac{Y^2}{n^2} \right ]$

$= E\left [ Y \right ] - \frac{E[Y^2]}{n}$

$= E\left [ Y \right ] - \frac{Var[Y]+(E[Y])^2}{n}$

$= np - \frac{np(1-p)+(np)^2}{n}$

$= np - p(1-p)- np^2$

$= np - p+ p^2- np^2$

$= (n - 1)p- p^2(n-1)$

$= (n - 1)(p - p^2)$

$= (n - 1)p(1-p)$

(b)

For unbiased estimator, $\tilde{V}$

$E[\tilde{V}] = np(1-p)$

If,

$\tilde{V} = \frac{n^2}{n-1}\left ( \frac{Y}{n} \right )\left (1- \frac{Y}{n} \right ) = \frac{n\hat{V}}{n-1}$

then,

$E[\tilde{V}] = E\left [ \frac{\hat{nV}}{n-1} \right ] = \frac{nE[\hat{V}]}{n-1} =\frac{n(n - 1)p(1-p)}{n-1} = np(1-p)$

Thus,

$\tilde{V} = \frac{n^2}{n-1}\left ( \frac{Y}{n} \right )\left (1- \frac{Y}{n} \right )$

is an unbiased estimator for V(Y).