Let X be a continuous random variable with the following PDF fx(x) = - 0, x 20 x< 0 where c is a positive constant. Find c.
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- Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)Let random variables X and Y have the joint pdf fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1 0, otherwise Find the joint pdf of U = X^2 and V = XY.Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. FInd (d) fZ|XY (z|x,y), (e) fX|YZ(x|y, z).
- Given the random variables X and Y having the following joint density: f(x, y) = 2(x + y) for 0 < y < x < 1. A) Compute the conditional pdfs: fX│Y (x) and fY│X (y). B) Are X and Y independent?Let X and Y be two continuous random variables having joint pdffX,Y (x, y) = (1 + XY)/4, −1 ≤x ≤1, −1 ≤y ≤1.Show that X ^2 and Y ^2 are independent.Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise c=1/2 P(X < 1), Determine whether X and Y are independent
- let X and Y be a random variables having pdf f(x,y)=2xy 0<x<y<1 Find P(X/Y<1/2)Use the moment generating function to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Poiss(λi), for i = 1, . . . , n.Find the distribution of Y = X1 + · · · + Xn.Suppose that a constant voltage source of 10 V supplies a current I (in mA) through a resistive load with a stochastic resistance R (in kilo ohms) defined by the PDF as follows. f_R (r)={█(1500r-750r^2-742.5, 0.9≤r≤1.1 0, elsewhere )┤ Since I = V/R, the derived distribution of I is defined by the following PDF: f_I (i)={█(1/i^4 (Ai+Bi^2+C), 100/11 ≤i≤ 100/9 0, elsewhere)┤ Where, A = ____, B = ____, and C = ____. Find the values for A, B, C, and give the expected value of I (in mA). (Express the answer in at least 4 decimal places)
- Let X be a continuous random variable with density function f(x) = 2x, 0 ≤ x ≤ 1. Find the moment-generating function of X, M(t), and verify that E(X) = M′(0) and that E(X2) = M′′(0).Let (X, Y ) be a random vector with joint density function of the form:f(X,Y )(x, y) = 24xy, 0 < x < 1, 0 < y < 1, x + y < 10 , otherwise(c)Compute the conditional probability P(X < 1/6|Y=1/2) and the conditional expectation E(X | Y = y)Suppose that the random variables X and Y have a joint density function f(x,y).prove that Cov(X,Y)=0 if E(X|Y=y) does not depend on y
