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Let X and Y be continuous random variables with the joint density function f(x,y) = 2, if 0 ≤ x ≤1,0 ≤ y ≤ 1,y ≤ x, and zero otherwise.
(a) Sketch the domain of the joint density function, that is, the support of (X, Y ). Then find the marginal density
(b) Compute E[X], E[Y ], V ar(X), V ar(Y ).
(c) Compute Cov(X, Y ) and ρ(X, Y ).Let X and Y be continuous random variables with the joint density function f(x,y) = 2, if 0 ≤ x ≤
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- 1) Let x be a uniform random variable in the interval (0, 1). Calculate the density function of probability of the random variable y where y = − ln x.Find 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.Let 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?
- Let X and Y be independent random variables with density f (x) = 3x² for 0 < x < 1. Then P (X+ Y < 1) is equal tob) Suppose that the joint density function of random variables X₁ and X₂ is given by f(x₁, x₂)= = 0, 0< x₁ < x₂ <2 otherwise i) Draw a sketch of the area where f(x₁, x₂) is not zero. ii) Find P(X₂ < 1). iii) Find E(X₁X₂).Let Y1 and Y2 be random variables with joint density functionf(y1, y2) = (6/7(y1^2+y1y2/2) 0 < y1 < 1, 0 < y2 < 2,0, elsewherea) Find marginal density functions. Are Y1 and Y2 independent?b) Find P(0 < Y1 < 0.3, −2 < Y2 < 1).c) Find P(0.6 < Y1 < 1|0 < Y2 < 1).
- Let X ∼ Unif(0, 1). Let g(x) = e x and Y = g(X). (i) Find the density fX of the random variable X. (ii) Find the cdf FY (y) = P(Y ≤ y) of Y . (iii) Find the density fY of the random variable Y .Let X be a continuous random variable with density f (x) = 24x-4 for x > 2. Then E (X) is equal toLet Y1 and Y2 be random variables with joint density function (yỉ + 2), 0 < y1 < 1,0 < y2 < 2, f(y1, Y2) elsewhere a) Find marginal density functions. Are Y1 and Y2 independent? b) Find P(0 < Y, < 0.3, –2 < Y2 < 1). c) Find P(0.6Random variables X and Y are independent and distributed as Uniform(−1,3).Find the density of random vector (Z,W), say f(z,w), where Z and W are defined asZ = X + 2Y, W = 2X −Y.Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 0Let (X,Y)' have density for r, y > 0, f (x, y) = (1+x)² (1+xy) > 0, otherwise. Show that X and X Y are independent, equidistriduted random variables and determine their distribution.SEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,