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- Let X have cdf Fx(x) = 1- (1+x)^-A, x,a > 0. Determine the pdf and cdf of Y = CXThe joint pdf of X and Y is given by fx,y(x,y) = {4x2 0<y<x<1} {0 otherwise} Let U =g(X,Y) = Y-X2. What is support of U?Find the linearization L(x, y) of f(x, y) = x2 y3 at (a, b) = (2, 1). Use it to estimate f(2.01, 1.02) and f(l.97, 1.01), and compare with values obtained using a calculator.
- Suppose the joint pdf of (X,Y) is f(x,y) =c(x2+y3), 0< x <1, 0< y <1, and f(x,y) = 0 otherwise. (a) Determine the value of c. (b) Find the conditional pdf ofXfor every given value ofY∈(0,1). (c) Compute Pr(X >1/2|Y= 1/2).Use the bordered Hessian to determine whether the stationary value of z is a maximum or minimum, z = x – 3y – xy, subject to x + y = 6Assume that X and Y have joint pdf f(x, y) ={x + 3x^2y if 0 <= x <= 1 and 0 <= y <= 1, 0 otherwise. (b) Compute the conditional pdf of X given Y = 1/4 . Simplify the final expression as much as possible.
- Show that the autocovariance function can be written asγ(s, t) = E[(xs − µs)(xt − µt)] = E(xsxt) − µsµt,where E[xt] = µtConsider the function f(x,y) = (3x^2 + bx)(2y^2+dy) where b and d are non-zero constants. Determine the values of b and d so that point (3,1) is a saddle point of f and calculate the value of b + d.Suppose that a person’s score X on a mathematics aptitude test is a number between 0 and 1 and that his score Y on a music aptitude test is also a number between 0 and 1. Suppose further that in the population of all college students in Belize, the scores X and Y redistributed according to the following joint pdf. f(x,y) = 2/5(2x + 3y) for 0≤x≤1, 0≤y≤1 and 0 otherwise What proportion of college students obtain a score greater than 0.8 on the mathematics test? If a student’s score on the music test is 0.3, what is the probability that his score on the mathematics test will be greater than 0.8?
- Let $X$ have the pdf $f(x)=4 x^{3}, 0<x<1,$ zero elsewhere. Find the cdf and the pdf of $Y=-\ln X^{4}$.p = m(1-dt) for d (buisness)Assume that X and Y have joint pdf f(x, y) ={x + 3x^2y if 0 <= x <= 1 and 0 <= y <= 1, 0 otherwise. (c) Compute Cov(X, Y ). You may use without proof the fact that E[X] = 17/24 and E[Y ] = 7/12 . Are X and Y independent? Justify .
