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- 5. Suppose that two continuous random variables X and Y have joint probability density function (e**y +e 2x-y) 15xs2,0sys3 elsewhere a. P( 3/2 s Xs2, 1SYS2) b. Are the random variables X and Y independent? c. find the conditional density X given Y = 0 d. Find the strength of the relationship and interpret the findings.6. Let f(x1,x2) = 1²e¬Mx1+*2), Let Y1 = X1 + X2 and X2 = Y2. (a) Find the joint distribution of Y1 and Y2: f(y1, y2) (b) Then compute the marginal density of Y1a) Let X₁, X₂, X3, X30 be a random sample of size 30 from a population distributed with the following probability density function: f(x)=2 if 02. Let X and Y denote independent random variables with respective probability density func- tions fx(x) = 2x, 0Let Ap and B be the random variables. A random process is defined a X () = Ag cos ont+ B sin ant, where og is a real constant. Find the power density spectru m of X (), if Ag and B, are uncorrelated random variables with zero mean and same variance.Let Y1, Y2, ..., Yn denote a random sample of size n from a population whose density is given by 3y2 0 < y< 0, f (y|0) = өз 0, elsewhere. 1.1. Show that Y(n) max(Y1, Y2, ... , Yn) is sufficient for 0.Choose a number B at random from the interval [0, 1] with uniform density. Find the probability that (a) 1/3 < B < 2/3(b) |B − 1/2| ≤ 1/4(c) B < 1/4 or 1 − B < 1/4(d) 3B2 < B4. Let X and Y be independent, continuous random variables with densities and fx(x) = = fy (y) = = 0 {} if 0 < x < 2 otherwise. y if 03. Let X be a scalar random sample from the following density f(x) = 2(8-x) 82 8-x Determine whether Q=- is a valid pivot. 0 for 0 ≤ x ≤ 0X (f) = A cos on i+ B sin wn t, where on is a real constant. Find the power Let An and B, be the random variables. A random process is defined as density spectrum of X (t), if A, and Bo are uncorrelated random variables with zero mean and same variance.6. Let X and Y be continuous random variables with joint density function 24xy if 0 < x < 1,0 < y < 1 – x f (x, y) = 0. otherwise. Calculate E(Y|X = }).4. Let X, Y be jointly continuous with density f(x, y) = = c(x - y)e-2 File Preview (a) Find the value of c. (b) Find the conditional distribution of X|Y. 0 < x<∞, -x < y < x otherwiseRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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