(a) What is the probability density function of aX + bY? (b) Compute E[X|X − 2Y] and E[X²|X – 2Y].
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![3. Let X, Y be independent, standard normal random variables.
(a) What is the probability density function of aX + bY?
(b) Compute E[X|X – 2Y] and E[X²|X – 2Y].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97e612ef-1556-436b-b62c-352b280e9e69%2F0d6e40e7-55b4-4b7c-9ac9-82ae5d634403%2Fclht7n7_processed.png&w=3840&q=75)
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- The random variables X and Y have a joint probability density function given by f(x, y) = way, 0 < x < 3 and 1 < y < x, and 0 otherwise.Q2 Let (X1, X₂) be jointly continuous with joint probability density function e-(x1+x2), 0 f(x1, x₂) = x₁ > 0, x₂ > 0 otherwise. Q2 (i.) Sketch(Shade) the support of (X₁, X₂). Q2 (ii.) Are X₁ and X₂ independent random variables? Justify your answer. Identify the random variables X₁ and X₂. Q2 (iii.) Let Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for Fy, (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and then find the probability density function of Y₁, i.e., fy, (y). 1 = Q2 (iv.) Let Mx, (t) = Mx₂ (t) (1 t), for t < 1. Find the moment generating function of Y₁, and using the moment generating function of Y₁, find E[Y₁]. Q2(v.) Let Y₂ = X₁ — X₂, and Mx₁ (t) = Mx₂(t) = (1 t). Find the moment generating function of Y2, and using the moment generating function of Y₂, find E[Y₂]. Q2 (vi.) Using the bivariate transformation method, find the joint…Let X and Y be independent random variables, prove that Var (XY) = Var (X) Var (Y) if E [X] = E[ Y] = 0.
- Let x be a random variable with probability density function p(x): 0.4(0.6)*- x=1,2,3, Then p(x 2 50) =4. Let X, Y be non-negative continuous random variables with probability density functions (pdf) gx(x) and gy (y), respectively. Further, let f(x, y) denote their joint pdf. We say that X and Y are independent if f(x, y) = 9x(x)hy (y) for all x, y ≥ 0. Further, we define the expectation of X to be E[X] = √rg(x)dx, to be the expectation of XY. 0 with a similar definition for Y but g replaced by h and x replaced by y. We also define E[XY] = (0,00)x (0,00) 110,00)x (0,00) 29 (x, y) dedy (0,∞) Use Fubini's theorem (which you may assume holds) to show that if X and Y are independent, then E[XY] = E[X]E[Y]. [2]Let X1, X2, and X3 be independent random variables from (-1, 1). Find the probability density function and the expected value of the random variable [X(1) + X(2)]/2.
- Exercise 20. Let X1 and X2 be iid U(0,1) random variables. Find the joint probability density function of Y1 = X1+ X2 and Y2 = X2 – X1.Let X and Y be two continuous random variables with joint probability density function f(x,y) = 2xy for 0 < x < y < 1. Find the covariance between X and Y.3. Show that the following are probability density functions: 1 (a) f(x) = -2*, x = 1, 2, ..., N, and zero elsewhere 2N+1-2 (b) f(x)=p(1-p)*, x=0, 1, 2, ..., and zero elsewhere; 0Let X and Y be independent random variables with joint probability density function fxy(x, y) = 1/3 (x + y), 0 < x <= 2, and 0 < y<= 1, and 0 otherwise. The marginal pdf fx(x) is given by O a. O b. O c. O d. (2 +2X)/3 (2 + 2X)/3 (X+1/2)/3 (X+1/2)/3 0 < X<= 2 0< X<= 1 0 < X<= 1 0Let X be a positive random variable having the probability density function fx(x) and 1 Y = be a decreasing function for 01)Let x be a random variable whose probability density function (pdf) is given by:and let A be the event {x >= 2}.a)Find E[x] , P[A] , px|A(X) and E [x | A].b) Let y=x2.Find E[y] and Var[y] .SEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,
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