2. Suppose the joint PDF of X and Y is given by - y²), 0≤x≤ 1, 0 ≤ y ≤ 1 ƒxy(x,y) = {c(x+y³²), otherwise (a) Determine c. (b) Find the probability that 0≤X≤1/4, 0≤Y≤ 1/4. (c) Determine the marginal PDFs fx(x) and fy(y).
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- 6. (Sec. 5.1) Two headlights of a car have the following joint pdf for their useful lifetimes X (the left headlight) and Y (the right headlight) ze(y+1) for r> 0.y > 0 f(x, y) 0 otherwise (a) What is the probability that the lifetime X of the left headlight exceeds 2.8? (b) Find the marginal pdfs of X and Y. Are the two lifetimes independent? Justify your answer (c) What is the probability that the lifetime of at least one headlight does not exceed 2.8?3. Suppose that X is a continuous random variable with pdf 3x², 0Find the marginal PDFs of X and Y.Let X and Y have the joint pdf f(x,y)= x+y , 0<=x<=1, 0<=y<=1. Find the marginal fx(x) and fy(y). Show if f(x,y) dependent. Compute mean(x), mean(y), variance (x) and variance(y)5. let X1 and X2 have the joint pdf f(x1,x2) = x1+ x2, 0 < x1 < 1, 0 < x2 < 1. Find the conditional mean and variance of X2 given X1 = x1, 0 < x1 < 1.Let X and Y be jointly continuous random variables with joint PDF given by: fx₁y (x,y) = = X (1+3y² ) / (0₁2) (X) I (01) (y) 4 a. Find the conditional PDF of X given y b. Find P ( ₁ < x < 1/2 | Y = £) 3 c. Find the marginal PDF of X.5. Let Y and Y have the joint pdf f(x, y) = 6(1 – x- y), x+ y < 1, 0 < x, 0 < y, zero elsewhere. Compute the marginal distributions as well as P(2X+3Y < 1) and E(XY+2X²) %3DQ9. Let X and Y have the joint pdf f(x,y) = 2, 0≤x≤ y ≤ 1. Which of the following is true? (A) The marginal pdf of X is fx(x) = 2(1-x),0 ≤ x ≤ 1. (B) The probability P(XQ9. Let X and Y have the joint pdf f(x, y) = 2, 0≤x≤ y ≤ 1. Which of the following is true? (A) The marginal pdf of X is fx(x) = 2(1-x),0 ≤ x ≤ 1. (B) The probability P(X < Y) = 1/2 (C) E(XY) = ² (D) The covariance Cov(X, Y) = 1/ (E) X and Y are uncorrelated.7. Let X be a continuous random variable with pdf f(x)=√ 0Consider the joint PDF f(x,y)=4xy for 0<x<1 and 0<y<1. (a) Are X and Y independent?(b) Find both marginal distributions. (c) Find both conditional distributions.Let Y1 < Y2 < Y3 < Y4 < Y5 denote the order statistics of a random sampleof size 5 from a distribution having pdf f(x) = e−x, 0 < x < ∞, zero elsewhere.Show that Z1 = Y2 and Z2 = Y4 − Y2 are independent.Hint: First find the joint pdf of Y2 and Y4.SEE MORE QUESTIONSRecommended textbooks for youAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage