Problem 4. Let X1, . . . , Xn be independent with common density f(x) = 2x 1[0 < x < 1]. Set Vn = max(X1, . . . , Xn).
. (b) Show that n(1 − Vn) → W in D holds for some random variable W and find the distribution function of W
Problem 4. Let X1, . . . , Xn be independent with common density f(x) =...
Let X1, . . . , Xn be independent with common density f(x) = 2x 1[0 < x < 1]. Set Vn = max(X1, . . . , Xn). (a) Verify Vn → 1 in P. (b) Show that n(1-Vn) → W in D holds for some random variable W and find the distribution function of W.
Problem 3. Let Xi,..., Xn be independent with common density 110 < æ < 1 Set Unmin(X,i,..., Xn). (1) Verify Un >o. (2) Show that n2Un U holds for some random variable U and find the distribution function of U
(7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi , . . . , X,.), V=min(X1, ,X,). (a) Find the distribution function and the density function of U and of V (b) Show that the joint density function of U and V is fe,y(u, u)= n(n-1)/(u)/(v)[F(v)-F(u)]n-1, ifu < u. (7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi...
4. Let X1,..., Xn be independent, identically distributed random vari- ables with common density 2 log c)? f(0; 1) = 0<<1, XCV21 (>0). : 212 (a) Find the form of the critical region C'* for the most powerful test of H:/= 1 vs. HQ: >1. (b) Suppose the n = 20 and a = .10. Find the specific value for the cutoff value) K from the critical region C* in part (a). (Hint: Show that Y = (log X/X) is...
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z and W, if it suppose N, X1, X2, are independent random variables and X,, have the same distribution function, F, and a) N-1 is a geometric random variable with parameter p (P(N-k), (k 1,2,.)) b) V - 1 is a Poisson random variable with...
Minimum and maximum of n independent exponentials. Let X1, X2, ..., Xn be independent, each with exponential (~) distribution. Let V min (X1, X2, ..., Xn) and W = max(X1, X2, ..., Xn). Find the joint density of V and W. .
Let X1, X2, ... be independent continuous random variables with a common distribution function F and density f. For k > 1, let Nk = min{n>k: Xn = kth largest of X1, ... , Xn} (a) Show Pr(Nx = n) = min-1),n>k. (b) Argue that fxx, (a) = f(x)+(a)k-( ++2)(F(x)* (c) Prove the following identity: al= (+*+ 2) (1 – a)', a € (0,1), # 22. i
4a). Let X1 and X2 be independent random variables with a common cumulative distribution function (i.e., c.d.f.) F(y) = { 0" if0cyotherwise。 Find the p.d. f. of X(2,-max(X, , xa). Are X(1)/X(2) and X(2) independent, where X(1,-min(X,, X2) ? 4a). Let X1 and X2 be independent random variables with a common cumulative distribution function (i.e., c.d.f.) F(y) = { 0" if0cyotherwise。 Find the p.d. f. of X(2,-max(X, , xa). Are X(1)/X(2) and X(2) independent, where X(1,-min(X,, X2) ?
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,.,Xn be independent random variables with common probability density f(x) = ה sin(x) , x E [0, π] (a) Assuming EX,] = 2, calculate Var(X). (b) Assuming Var(Xs + + X,) = Var(X) + Var(Xn) and if a, b є R that Var(ax, + b) = a2Var(X), calculate the mean and the variance of Zn, defined [1 as follows: Var(X1+...+ Xn)