Let the joint density function of (X,Y) be: f(x,y) = (x+y) / 4, 0 < x < y < 2. a) Find fX|Y(x|y). b) Compute the conditional probabilities P(X < 1/2 | Y = 1) and P(X < 3/2
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Let the joint density
f(x,y) = (x+y) / 4, 0 < x < y < 2.
a) Find fX|Y(x|y).
b) Compute the conditional probabilities P(X < 1/2 | Y = 1) and P(X < 3/2 | Y = 1).
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- Assume x and y are functions of t. Evaluate dydtfor each of the following. x2+yxy=9; dxdt=2,x=4,y=2Q1) Discrete joint variables X and Y with probability density f(x,y) (pdf) are given in this table. Find: 1) The Covariance Cov(X,Y)? (Cov(X,Y) = MxY-MxMY) 2) The correlation (pxy) between X and Y where Pxy = Cov(X,Y) PXPY Y 3 fx(x) f(x,y) 1 2 1 1/4 1/4 0 X 2 0 1/4 1/4 fy(y) Note that 2 n=2 n=3 Px² = Σn²±²x² f(x, y) - μ and py² = Σ3y² f(x, y) – µ Zk=0Let X and Y be two continuous random variables with joint probability density function: 6e-(2x+3y)+k x21, y > 0 fxy (x, y) = otherwise a) Find the coefficient k. b) Find P(X 2). c) Find the marginal probability density functions of X and Y ( fx(x), f;(y)). d) Are X and Y independent?
- c) Let X [0,1]x[0,1]x[0,1]. Let Y = g(X), be a two dimensional random vector, where 91(X ) = X1 + X2 , g2(X) = X2 – X3 . Find the density function of Y. [X1,X2, X3] be a three dimensional random vector uniformly distributed onDetermine k so thatf(x, y) = kx(x − y) for 0 < x < 1, −x < y < x0 elsewherecan serve as a joint probability density.1)Let x be a uniform random variable over the interval (0, 1). Knowing that y = x2 , calculate:a)Determine Fy(Y) = P(y<=Y),Y real and determine the pdf of y.b)Calculate E[x2] , using the pdf of x.c)Calculate E[y], using the pdf of y and compare with part (b).
