4.3-8 (a) Find a constant b (in terms of a) so that the functionomit o ovi tbe be x+y) 0<x<a and Assum Problem 0...
Find a constant k (in terms of a) so that the function fxx (x,y) = e-(x+u) 0 << oo and 0 < y <a and O elsewhere is a valid joint density function.
4.3-17. Find the marginal densities of X and Y using the joint density Sx.x(x,y) = 2u(x)u(v)exp- anos a b 4.3-19. The joint density of two random variables X and Y is E fx.y(x,y) = 0.18(x)8(y)+0.128(x - 4)8(y) Problem +0.058(x)(y-1) +0.258(x-2)(y-1) valaltitude +0.38(x-2)8(y - 3) +0.188(x-4)8(y - 3) Find and plot the marginal distributions of X and I.
Problem 1: Given the function g(x,y)-ke-xy-c)u(x-a)u(y-b) find the constant "k" in terms ofa, b, and c so that g(x,y) is a valid probability density function (15%). Are the random variables X and Y statistically independent (10%)? (Support your answer.)
Problem 5 20 marks total 0 < y < x2 < Consider two rvs X and Y with joint pdf f(x,y) = k-y, Sketch the region in two dimensions whereAx.y is positive. Then find the constant k and sketch /fx.y) in three dimensions. I4 marksl (a) Find and sketch the marginal pdf/(x), the conditional pdf ffr11/2), and the conditional cdf F11/2) (b) 4 marks] Find P(XcYlY> 1/2), E(XIY=1/2) and E(XlY>1/ 2). /4 marks/ (c) (d) What is the correlation between...
. For > 0 and A > 0, define the joint pdf -Ay = 0<x<A,<y, fx.y(,y) 10 else. (a) Express c in terms of X and A. (b) Find E[XY]. (c) Let [2] be the largest integer less than or equal to z. For example, (3.2] = 3 and [2] = 2. Find the probability that [Y] is even, given that 4 <x< 34
4. Let X and Y have joint density function le-x 0 < y < x < 0 Jxy(x, y) = lo elsewhere Another random variable of interest is U=X–Y. Find the probability density function for U.
2. Let X and Y be two continuous random variables varying in accordance with the joint density function, fx.y(z, y-e(x + y) for 0 < z < y < 1. Solve the following problem s. (1) Find e, fx(a) and fy (v) (2) Find fx-u(z) and fY1Xux(y) (8) Find P(Y e (1/2, 1)|X -1/3) and P(Y e (1/2,2)| X 1/3). 3. Find P(X < 2Y) if fx.y(zw) = x + U for X and Y each defined over the unit...
Problem 4. Find a function v(x) so that the substitution y(x) = u(x)+(x) transforms the differential equation y" + P(x)y' + Q(x)y = 0 into an equation for u of the form u" + f(x)u = 0. Write the function f in terms of P and Q. The last equation is called the normal form of a homogeneous linear second order equation.
1. Let X and Y have a discrete joint distribution with ( P(X = x, Y = y) = {1, 10, if (x, y) = (-1,1) if x = y = 0 elsewhere Show that X and Y are uncorrelated but not independent. [5 points] 2. Let X and Y have a discrete joint distribution with f(-1,0) = 0, f(-1,1) = 1/4, f(0,0) = 1/6, f(0, 1) = 0, $(1,0) = 1/12, f(1,1) = 1/2. Show that (a) the two...
) Let X, Y be two random variables with the following properties. Y had density function fY (y) = 3y 2 for 0 < y < 1 and zero elsewhere. For 0 < y < 1, given Y = y, X had conditional density function fX|Y (x | y) = 2x y 2 for 0 < x < y and zero elsewhere. (a) Find the joint density function fX,Y . Be precise about where the values (x, y) are non-zero....