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- e) The density function of a certain random variable X is given by xa-'(1– x)®-1 f(x) = {B(a,0) 0, if 0 0. i) Derive the expected value of X. ii) Derive the variance of X.Let X1, X2,... , Xn be independent Exp(A) random variables. Let Y = X(1)min{X1, X2, ... , Xn}. Show that Y follows Exp(nA) dis- tribution. Hint: Find the pdf of Y2. Let X and Y be jointly continuous random variables with joint PDF x + cy2 fy(X.v) = OsxS1,0Sys1 elsewhere la) Find the constant c b) Find the marginal PDF's fx(x) and fy(y) c) Find P(OSXS1/2,05YS/2)
- Let X and Y be jointly continuous random variables with joint PDF cx +1, 2,y> 0, x+ y<1 0, fx,x (x, y) = otherwise. Then the constant c= And the probability that P(Y < 2X²) = ;: Here a = and b =2. Let X and Y be jointly continuous random variables with joint PDF x + cy2 0, OSXS1,0Sys1 elsewhere a) Find the constant c Find the marginal PDF's fy(x) and fy(y) c) Find P(OSXS1/2,0SYS1/2) b)2. Let X and Y be jointly continuous random variables with joint PDF x + cy2 0, OSXS1,03. If X and Y are jointly continuous random variables with joint PDF fx.y (r, y) = exp(-y)I0.v)(x)I(0,0) (y) (a) Find the joint MGF of X and Y (b) Find the marginal MGFS of X and Y (c) Identify the marginal distribution of X and Y (d) Find the E(XY) using the joint MGF of X and YWe have a random variable X and Y that jave the joint pdf f(x) = {1 0<x<1, 0<y<1} {0 otherwise} If U = Y-X2 , what is the support for the random variable U? What would fu(u) and Fu(u) be? Say U = Y/X, what is the support for the random variable U? What would fu(u) and Fu(u) be?Let X and Y be two jointly continuous random variables. Let Z = X² + Y². Show that F2(=) = L Fxr(V=- yř 19) – Fxy(-v/= = y"l»)] fv (w) dy .A random variable Y have a distribution with parameter a > 0 and yo > 0 if its density function is given by: ´ay§ f(y) = {ya+i if y > yo (0, elsewhere i) Derive the mean of Y. Show all necessary steps. ii) ayg (a-2)(a-1)2" Show whether or not that the variance of Y is3. Let X and Y be continuous random variables with joint PDF (3x 0s ysxs1 f(x, y) = {* otherwise Determine the correlation of variables X and Y.Let X and Y be two continuous random variables with joint probability density [3x function given by: f(x,y)= 0SEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage