Due before class on June 5th, 2012, Tuesday. Please show your steps, answer conc

Question

Due before class on June 5th, 2012, Tuesday. Please show your steps, answer concisely and write clearly. You may work in groups, but each member of a group should submit their own write-ups. In the case of group collaboration, you are required to write down the names of the people you work with. The problem set has a total of 10 points. X is a discrete random variable which takes values {-2, -1, 1, 2} with equal probability. That is, Pr(X = -2) = Pr(X = -1) = Pr(X = 1) = Pr(X = 2) = 1/4. Calculate E(X) and E(X2). Calculate Cov(X; X2). Are X and X2 independent? Why or why not? (You can explain in words. Be concise! X is a discrete random variable and takes only three values. Its probability density distribution (pdf) is given by Y is a continuous random variable whose conditional expectation given X can be expressed by function h(X) such that E(Y|X) = X2 - X Compute E(Y) using the Law of Iterated Expectations (E(Y) = E[E(Y|X)]). Conditional variance of Y given X, V ar(Y|X) is defined by V ar(Y|X) = E[(Y - mu Y|X)2|X], where mu Y|X = E(Y|X). It is known that Var(Y|X) = E(Y2|X) - (E(Y|X))2. If E(Y2|X = 1) = 1, calculate Var(Y|X = 1). Let {Y1, Y2, Y3, Y4} be four indepent, identically distributed (i.i.d.) random variables from a population with mean mu and variance sigma 2. Consider the following estimator of Prove that is an unbiased estimator of mu. Express Var in terms of sigma2. Suppose you know that mu = 0. Is the estimator Ohm as defined below an unbiased estimator of sigma 2? Show you work. (Hint V ar(Y) = E(Y2) - (E(Y))2.) Let F(x) be the cumulative distribution function (cdf) of a t distribution with 9 degrees of freedom. It is known that F(2.262) = 0.975 and F(1.833) = 0.95. Let {Y1, Y2, ... Y10} be a random sample of 10 drawn from normal distribution with unknown mean mu and unknown variance sigma2. Define and S as Using as an estimator of mu, write down an expression of the test statistics using and S. What distribution does the test statistics follow under the null hypothesis? We obtain from a sample that and S = 3. calculate the sample test statistics for mu = 1. Consider the following hypothesis test with two-side alternative H0 : mu = 1 What is the decision rule for the two-sided test above with 5% signicance level? (Please specify the critical value.) Carry out the one-sided test below using the test statistics from (c). H0 : mu = 0 H1: mu > 0

Explanation / Answer

Answer 4.
i)When the null hypothesis is Ho:=0, the test statistic is:

t={n(Y bar - 0)}/S with (n-1) degress of freedom

When the null hypothesis is H0:=0, the test statistic is:

Chi Square=(Yi-Ybar)2/02=(n-1)S2/02 with (n-1) degrees of freedom

ii) the test statistic follows a t distribution when  Ho:=0

the test statistic follows a chi-square distribution when H0:=0

iii)t==10(5-1)/3=3.16*4/3=12.64/3=4.2133

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