Liniar combinations of random variables 1. Let X and Y be two continuous random variables with joint density function: a. Calculate P(X+Y ≥ !) b. Calculate E(Y-X) c. Calculate Var (Y - X). f(x,y)= 0
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Q: i) Draw a sketch of the area where f(x₁, x₂) is not zero. ii) Find P(X₂ < 1). iii) Find E(X₁X₂).
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A: Solution: From the given information, joint density function of X and Y is
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Q: a) Find marginal density functions. Are Y1 and Y, independent? b) Find P(0 < Y < 0.3, –2 < Y2 < 1).…
A: given f(y1,y2) = 6(y12 + y1 . y2/2) /7 , 0<y1<1 ,0<y2<2
Q: i) Draw a sketch of the area where f(x₁, x₂) is not zero. ii) Find P(X₂ < 1). iii) Find E(X₁X₂).
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Q: 2. The continuous random variables X and Y have known joint probability density function 4x 0≤ y…
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Q: E (X+ Y)
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Q: Show E(Y)
A: First we have to determine the marginal pdf of Y to calculate the value of E(Y).
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- Let X and Y be two random variables with joint density function f(x,y) = (3 − x + 2y) / 60, for 1 < x < 3, 0 < y < 5. Is P(X > 2, Y < 3) equal to P(X > 2) × P(Y < 3)?The multidimensional random variables H= (c) Let x := 2731 W 21 X2 Yı = follows N₁(μ, E) where 7 3 Y 2 3 Y 6 0 0 5 4 -2 2 4 -2 G (a) Determine all the independent univariate random variables. (b) What is the probability density function for x₁ + 2x₂? [2] tion density function p(y|x) and p(x|y = 1). and y := [₁]. Determine the conditional distribu-2. Let X and Y denote independent random variables with respective probability density func- tions fx(x) = 2x, 0Show V(Y)Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 04. Let X and Y be independent, continuous random variables with densities and fx(x) = = fy (y) = = 0 {} if 0 < x < 2 otherwise. y if 03. Let random variables X and Y be independent with joint density fe(x, y). Let Ix (0) and Iy (0) be the Fisher information of X and Y, respectively. Prove that the Fisher information I(x,y)(0) = Ix(0) + Iy(0).Please helpFind the density of U = Y1 +Y2, where Y1 and Y2 are independent random variables with densities (yı – 1), 3 < Yı < 4, (y2 + 2), 0 < Y2 < 1, fy, (y1) : fy,(y2) = otherwise, otherwise.4) The joint density function of the random variables X and Y is given by (&xy f(x, y) = 0SXS1,0 sysx otherwise Find (a) the marginal density of X, (b) the marginal density of Y, (c) the conditional density of X, (d) the conditional density of Y.6. Let X and Y be continuous random variables with joint density function 24xy if 0 < x < 1,0 < y < 1 – x f (x, y) = 0. otherwise. Calculate E(Y|X = }).a) What is the value of k?Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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