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- 5. Let f(x1,x2) =21x²x², 0< x1 < x2 < 1, zero elsewhere be the joint pdf of X1 and X2. (a) Find the conditional mean and variance of X1, given X2 = x2, 0 < x2 < 1 (b) Find the distribution of Y = E(X¡|X2) %3D (c) Determine E(Y) and Var(Y) and compare these to E(X1) and Var(X1), respectively.Find a number c that satisfies the conclusion of the Mean Value Theorem for f(x) = x +x – 1 on the interval [0, 2] %3D V3 1 2. V3 V3 2. O V312. Please do all the development
- 6State the CLT for independent identically distributed real RVs with mean u and variance a². Suppose that X. X,. . . are independent identically distributed random variables each uniformly distributed over the interval [0, 1]. Calculate the mean and variance of In X. Suppose that 02. A random variable X has a p.d.f. f(x) given by √(1-x)6 0d,eLet X be a random variable. Suppose we try to use a constant real number â to guess the value of X. Let E(X – î)² be the mean squared error (MSE) of this estimation. (i) By writing X − â = (X − E X) + (EX − 2), show that E(X)² = Var(X) + (EX - 2)². = (ii) Conclude that the MSE is minimized for î Var(X). EX and the minimum MSE equals toLet X; € {1,2, 3, ...} be the number of days until relapse for patient i who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf iid X1, X2,..., Xn fxp(x | p) = (1 – p)*-'p for 0 < p < 1 defined on x E {1,2, 3, ...} and 0 elsewhere. Here, p is the risk of relapse on each day. 1. Using a p~ Beta(a, B) prior, derive the posterior density of p, fp|X, (p | Xn).3. Let the random variable X have the pdf f(x) = 2(1 — x), 0 ≤ x ≤ 1, zero elsewhere. a) Find the cdf of X. Provide F(x) for all real numbers x (set up the appropriate cases). b) Find P(1/4 < X < 3/4). c) Find P(X= 3/4). d) Find P(X ≥ 3/4).Let X be a random variable on a closed and bounded interval [a, b]. Let g(x) be a convex function. Prove that g(E(X)) ≤ E (g(X)Let X„ X„ ...... X, be a random sample from a distribution 2 with p.d.f. given by : f (x, 0)= e- (x – 0), 0SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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