12. Let the random variable X and Y have joint pdf4f(x,y) = (x² + 3y²), 0

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Suppose that that the RV X has the Gamma(2,β ) distribution for some β > = 0, E(X) = 2β and V (X) = 2β. Further, suppose that given X = x , the RV Y has the Unif (0, x) distribution. a) Write down the conditional pdf of Y given X = x (f_Y|X )(y | x) = b) Find the joint pdf of (X,Y) (f_XY)(x,y)= c) Find the conditional mean and variance of Y E(Y | X=x) =____________ V(Y |X=x )= ____________ d) Find the unconditional mean and variance of Y E( Y)= ____________ V(Y ) =____________ e) Find the marginal pdf of Y (f_y)( y)= ___________ f) Identify the distribution of Y Y-_________ g) Use parts e)- f) to find mean and variance of Y E(Y )=____________ V( Y)= ___________ The answer for Is 1/x (0<y<x) Β2 x/2 , (x2)/2 β , β please solve e) and f) and g)
3. Let X, Y be continuous random variables with joint pdf f(x, y). Calculate the joint cumulative distribution function of (X, Y) for each of the pdfs below. (a) f(x, y) = 6e-3x-2y, x, y ≥ 0. (b) f(x, y) = x + ³y²,0 ≤ x, y ≤ 1.
Let X and Y be discrete random variables with joint pdf f(x,y) given by the following table: y = 1 y = 2 y = 3 x = 1 0.1 0.2 0 x = 2 0 0.167 0.4 x = 3 0.067 0.022 0.033 Find the marginal pdf’s of X and Y. Are X and Y independent?
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).
Suppose X has PDF fX (x) = 4x(1 − x^2) for 0 < x < 1 (0 everywhere else). Find all mediansand modes for X.(a) Find all medians of X.(b) Find all modes of X.
Show Corr(X, Y)
2. Let X and Y be jointly continuous random variables with joint PDF x + cy2 0, OSXS1,0Sys1 elsewhere a) Find the constant c Find the marginal PDF's fy(x) and fy(y) c) Find P(OSXS1/2,0SYS1/2) b)
(b) Let X1, X2, ...,X, be a random sample from the pdf Be z for z 2 0 f(z,0) = 0, otherwise (1) Show that the statistic T(X1, X2, . . , Xn) = X, is complete sufficient for 8. %3D i=1
3. Let X; N iid U(0, 1) for i e {1, 2, ...,n}. (a) Compute the pdf of Y; = - log(X;), name the distribution that Y; follows and provide any parameter value or values needed. (b) Compute the pdf that W = [I", X, = X1 X2... X, follows. You may use the result given as part of the Example 2.6 video (even though it will only be proved in Chapter 3). (c) Compute E[W] in the two following ways: (i) through exploiting independence of the X, and using EUV] = E[U]E[V] for any independent random variables U and V. (ii) using the pdf of W computed in the previous part.
Assume that X1, X2, X3 ∼ Exp (λ) are independent and evenly distributed with the distribution Exp (λ = 1). (a) Determine the distribution of Y = X1 + X2 + X3 and state its PDF f (y). (b) Determine the distribution of U = 2Y
(b) Let X represent the lifetime in months of a certain brand of light bulb. The pdf of X is given by f(x)=K(6-x), 0
Let X, and X; be independent exponential random variables having pdfs f, (x) = e, x > 0 and fr,(x2) = e"", I3 > 0. Given that Y, = X,/X and Y, = X, + X2. a. Find the joint pdf of Y, and Y2 b. Find the marginal pdf of Y1.

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12. Let the random variable X and Y have joint pdf f(x,y) = (x² + 3y²), 0