Zi= 1 if ith primary unit is in the sample, and Zi=0 otherwise
Zi= 1 if ith primary unit is in the sample, and Zi=0 otherwise proof: Var (&...
Let Zi, Z.Zg be a random sample of size 3 from the N(μ = 0, σ2-1) distribution. Let Xi, X2 be a random sample of size 2 from the N( 1-0,02-2) distribution. Let Y.Y2, Y be a random sample of size 3 from the N(11-1,ơ2-3) distribution. The Xi, Y, and Zi are all mutually independent. Give the distribution (including parameters) of each of the following: 2
2. Let Xi,... Xn be a random sample from the density f(x:0) 1o otherwise Suppose n = 2m+1 for some integer m. Let Y be the sample median and Z = (a) Apply the usual formula for the density of an order statistic to show the density max(X1) be the sample maximum. of Y is 0) 6 3) (b) Note that a beta random variable X has density re+ β22 a-1 (1-2)8-1 with mean μ α/G + β) and variance...
4.4.19 Random variableX has PDE fx(a)-1/4 -1s-33, 0 otherwise Define the random variable Y by Y = h(X)X2. (a) Find E[X and VarX (b) Find h(E[X]) and Eh(X) (c) Find ElY and Var[Y .4.6 The cumulative distribution func- tion of random variable V is 0 Fv(v)v5)/144-5<7, v> 7. (a) What are EV) and Var(V)? (b) What is EIV? 4.5.4 Y is an exponential random variable with variance Var(Y) 25. (a) What is the PDF of Y? (b) What is EY...
4. Given the pd )if -1,2 0 otherwise. (a) Determine the proper value for c. (b) Find P(0 < X<1) (c) Find E(X) (d) Find E(X) and Var(X) 4. Given the pd )if -1,2 0 otherwise. (a) Determine the proper value for c. (b) Find P(0
Consider the multiple regression model y = X3 + €, with E(€)=0 and var(€)=oʻI. Problem 1 Gauss-Mrkov theorem (revisited). We already know that E = B and var() = '(X'X)". Consider now another unbiased estimator of 3, say b = AY. Since we are assuming that b is unbiased we reach the conclusion that AX = I (why?). The Gauss-Markov theorem claims that var(b) - var() is positive semi-definite which asks that we investigate q' var(b) - var() q. Show...
1. Consider the joint probability density function 0<x<y, 0<y<1, fx.x(x, y) = 0, otherwise. (a) Find the marginal probability density function of Y and identify its distribution. (5 marks (b) Find the conditional probability density function of X given Y=y and hence find the mean and variance of X conditional on Y=y. [7 marks] (c) Use iterated expectation to find the expected value of X [5 marks (d) Use E(XY) and var(XY) from (b) above to find the variance of...
(proof) n all 26. Let P(z) = 0 stand for an the zeros of which are in the unit circle |z| < 1. Replacing each coefficient of P() by its conjugate we obtain the polynomial P(2). We define p*()=P( The roots of the equation P(z) + P*(2) = 0 are all on the unit circle |z| = 1 algebraic equation of degree n all 26. Let P(z) = 0 stand for an the zeros of which are in the unit...
For all of the following, consider the joint pdf f(, y) c(3ry) for a, yE (0, 1 and 0 otherwise. 5. Given a random sample of 100 observations from X, find P(.5 < X HINT: Use the CLT .55) For all of the following, consider the joint pdf f(, y) c(3ry) for a, yE (0, 1 and 0 otherwise. 5. Given a random sample of 100 observations from X, find P(.5
suppose that ri 1, are a random sample having probability density function f(x;8)=(0 otherwise (a) Determine the method of moments estimator of δ based on the first moment.
Suppose that Case i is an outlier in the sense that an amount δ is added to its expected value. Let u i be the unit vector with 1 in the ith position and zeros elsewhere and write the revised model is y = Xβ + δu i + e. Weisberg (1985) calls this the mean shift outlier model. Using standard methodology, develop the test statistic for the hypothesis, H 0 : δ = 0 and show that the resulting...