find the extreme values of f
f(x1,x2,...,xn) = x1x2...xn(1 − x1 − 2x2 − ··· − nxn), if x1 > 0, x2 > 0, ...,xn > 0.
Find the extreme values of f f(x1,x2,...,xn) = x1x2...xn(1 − x1 − 2x2 − ··· − nxn), if x1 > 0,...
Find the optimal bundle for the following utility functions and for budget line (P1X1+P2X2=m) a) U(X1,X2)=X1X2 b) U(X1,X2)=X1^2X2^3 c) U(X1,X2)=X1^2+2X2 d)U(X1,X2)= ln (x1^3X2^4) e) U(X1,X2)= 2X1+X2 f) U(X1,X2)= min (2X1,X2)
Problem 1. Simplify the logic expression using Boolean Algebra. f(x1 ,x2, x3) = x1'x2'x3' + x1x2'x3' + x1'x2'x3 + x1x2x3 + x1x2'x3 Problem 2. Simplify the logic expression given in problem 1 using K map.
2.1.1. Let f(x1,x2) = 4x1x2 , 0 < 띠 < 1, 0 < x2 < 1, zero elsewhere, be the pdf of Xi and X2. Find P(0 < Xìく, ¼ < X2 < 1), P(Xi = X2), P(Xi < X2), and Hint: Recall that P(X1 -X2) would be the volume under the surface f(xi, r2)- 4 t 0 < x1 = x2 < 1 in the x1x2-plane. T102 and above the ne segmen
3. Let (X1, X2) have the joint p.d.f 1 if 0 <1,0 < <1 f(1, ) else Find P(X1X2 < 0.5)
Given random variables X1, X2, Y with E[Y | X1, X2] = 5X1 + X1X2 and E[Y 2 | X1, X2] = 25X2 1X2 2 + 15, find E[(X1Y + X2) 2 | X1, X2]. ㄨ竺Bin(2.1/4). Suppose X and Y are independent random variables. Find the expected value of YX. Hnt: Consider conditioning on the events (X-j)oj0,1,2. 9. Given random variables XI,X2, Y with E'Y | XiN;|-5X1 + X1X2 and Ep2 1 X1,X2] 25XX15, find 10. Let X and Y...
Let X1, X2,..., Xn be a r.s. from f(x) = 0x0-1, for 0 < x <1,0 < a < 0o. (a) Find the MLE of 0. (b) Let T = -log X. Find the pdf of T. (c) Find the pdf of Y = DIT: (i.e., distribution of Y = - , log Xi). (d) Find E(). (e) Find E( ). (f) Show that the variance of 0 MLE → as n → 00. (g) Find the MME of 0.
Let X1 and X2 have joint PDF f(x1,x2)=x1+x2 for 0 <x1 <1 and 0<x2 <1.(a) Find the covariance and correlation of X1 and X2. (b) Find the conditional mean and conditional variance of X1 given X2 = x2.
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...
Problem 2. (The Convergence of Extreme Value) Let X1, X2, ... be i.i.d sample from the distribution with density function as: f(x) = >1 10 otherwise Define Mn = min(X1, X2, ... , Xn), answer the following questions. 1) Show that Mn P 1 as n +0. 2) Show that n(Mn – 1) converges in distribution as n + 00. Find out the limit distri- bution.
(Unconstrained Optimization-Two Variables) Consider the function: f(x1, x2) = 4x1x2 − (x1)2x2 − x1(x2)2 Find a local maximum. Note that you should find 4 points that satisfy First Order Condition for maximization, but only one of them satisfies Second Order Condition for maximization.