6. Suppose X and Y are random variables with joint density given by 32 5(x² + y²) if 0≤x≤ 2,0 ≤ y ≤2 if else p(x, y) = 0 Compute the following probabilities a. P(X + Y ≤ 2) b. P(X² + Y² ≤ 4) (Hint: use polar coord.)
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- Let X ~ N(0, 2) and Y ~ covariances Cov(X,Y) and Cov(Y, X +Y). Exp(A = 3) be two uncorrelated random variables. Find theQ2) A continuous random variable has PDF Kx²+2x+1, -25x5 3. Find K, P(x)>0. X, X² and ¹.Suppose that the random variables X and Y are independent with Var(X)=8 and Var(Y)=6. Calculate Var(5X−7Y+17)
- Is cov(X, Y) random?5. Let X1 - b(n1, p) and X2 ~ b(n2,p) be independent random variables. a) Find the m.g.f. of Y = X1+ X2. %3D b) How is Y distributed?Let X Binom(16, 14 ) and Y Geom( 12 ) be two independent random variables. Compute(i) E(X + 2Y + 3),(ii) the standard deviation of 3X + 5,(iii) V ar(2Y + 1),(iv) Cov(2X, 3Y ),(v) Cov(X, 4X).
- Suppose X is random variable whose p.d.f. is f2)=(2x-x²), 0, OSxs2 Suppose X is random variable whose p.d.f. is f(x)={4 elsewhere Find the mode , if it exists.Q1. Suppose that X and Y arejoirtly continuDus random variable for 0Suppose we have the quadratic function f(x)=A(x^2)+2X+C where the random variables A and C have densities fA(x)=(x/2) for 0≤x≤2, and fC(x)=3(x^2) for 0≤x≤1. Assume A and C are independent. Find the probability that f(x) has real rooX and Y are two discrete random variables. If the covariance between X and Y is positive, then: of + of_ož-y. A) В) X and Y are two discrete random variables. If X and Y are positively correlated, then: ož + oỷ. 6. A) В) V A 5.Suppose that X and Y are random variables with E(X) 2 , E(Y) = 5 and E(x?) = 8, E(Y?) = 30 and cov(2X – Y,4X – 3Y) = -12, then cov(-5X – 3,2Y + 4) is equal to: Hint: cov(ax + bY,cX + dY) = acV(X) + bdV(Y) + (bc + ad)cov(X,Y) O -120 O -59 O -64 O -30Let X and Y be random variables having the same distribution. Show that Cov(X +Y, X – Y) = 0.SEE MORE QUESTIONS