Question 4 Let X ~ N(4, 02) (a) Use the properties of a Normal distribution to...
proof for distribution of (n-1)S^2/sigma^2 is the chi square distribution with n-1 degrees of freedom. I don't understand the expansion of the square, specifically how certain terms disappeared and how a sqrt(n) appeared. Also towards the end, why does V have a degree of freedom of 1? x A detailed explanation of what happened from step 2 to step 3 would be very helpful! THEOREM B The distribution of (n − 1)S2/02 is the chi-square distribution with n – 1...
1. The Weibull distribution has many applications in reliability engineering, survival analysis, and general insurance. Let β > 0, δ > 0. Consider the probability density function x>0 zero otherwise. Find the probability distribution of W-X c Determine the probability distribution of W by finding the m.g.f. of W, Mw(t) Find the mgf. of w. Mw(t)-E(e, w )-E(e'x®). 1: u-substitution:v5 t X i) Hint -substitution:-»5. Hint', 2: Must have t< for the integral to converge . i What is the...
8. An important distribution in the multivariate setting is the multivariate normal distribution. Let X be a random vector in Rk. That is Xk with X1, X2, ..., xk random variables. If X has a multivariate normal distribution, then its joint pdf is given by f(x) = {27}</2(det 2)1/2 exp {=} (x – u)?g="(x-1)} is the covariant matrix. Note with parameters u, a vector in R", and , a matrix in Rkxk that det is the determinant of matrix ....
3, Let X, X2,X, be independent random variables such that Xi~N(?) a. Find the distribution of Y= a1X1+azX2+ i.(Hint: The MGF of Xi is Mx, (t) et+(1/2)t) + anXn +b where a, 0 for at least one b. Assume = 2 =n= u and of- a= (X-)/(0/n) ? Explain. a. What is the distribution of The Sqve o tubat num c. What is the distribution of [(X-4)/(0/Vm? Explain.
Let X1,.. ,X be a random sample from an N(p,02) distribution, where both and o are unknown. You will use the following facts for this ques- tion: Fact 1: The N(u,) pdf is J(rp. σ)- exp Fact 2 If X,x, is a random sample from a distribution with pdf of the form I-8, f( 0,0) = for specified fo, then we call and 82 > 0 location-scale parameters and (6,-0)/ is a pivotal quantity for 8, where 6, and ô,...
Only 1-4) X, be a random sample from N(4,a ), , and let X and S be sample mean and sample 1. Let variance, respectively. In Order to show that and S are independent, tollow the steps below. and show the joint pdf of X,X3,*, X 1-1) Use the change of variable technique = Nx = x - is (п-1)5? п(т-и? f(E,x,) еxp ov2x 2a2 Use Jacobian for n x n variable transformation 1-2) Use the fact that X~N(u,a n)...
4. Let (X,Y) be a bivariate normal random vector with distribution N(u, 2) where -=[ 5 ], = [11] Here -1 <p<1. (a) What is P(X > Y)? (b) Is there a constant c such that X and X +cY are independent?
1. Let X be an iid sample of size n from a continuous distribution with mean /i, variance a2 and such that Xi e [0, 1] for all i e {1,...,n}. Let X = average. For a E (0,1), we wish to obtain a number q > 0 such that: (1/n) Xi be the sample Р(X € |и — 9. и + q) predict with probability approximately In other words, we wish to sample of size n, the average X...
Let X1,X be a random sample from an EXP(0) distribution (0 > 0) You will use the following facts for this question: Fact 1: If X EXP(0) then 2X/0~x(2). Fact 2: If V V, are a random sample from a x2(k) distribution then V V (nk) (a) Suppose that we wish to test Ho : 0 against H : 0 = 0, where 01 is specified and 0, > Oo. Show that the likelihood ratio statistic AE, O0,0)f(E)/ f (x;0,)...
1. Let X1, ..., Xn be a random sample of size n from a normal distribution, X; ~ N(M, 02), and define U = 21-1 X; and W = 2-1 X?. (a) Find a statistic that is a function of U and W and unbiased for the parameter 0 = 2u – 502. (b) Find a statistic that is unbiased for o? + up. (c) Let c be a constant, and define Yi = 1 if Xi < c and...