Problem #7: The joint probability mass function of (X, Y, W) is Problem #7(a): Problem #7(b): p(1, 2, 3) = p(2, 1, 1) = 0.4, p(2, 2, 1) = p(2, 3, 2) = 0.1. (a) Find E[XYW]. (b) Find E[XY+YW].
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- Problem #2: Let X be a continuous random variable with probability density function 16xe 4x x > 0 o Problem #2(a): Problem #2(b): f(x) = otherwise (a) Find the value of the moment generating function of X at t = 1. (b) Find E[X]. Enter your answer symbolically, as in these examples Enter your answer symbolically, as in these examplesJ 1 Problem 126. Let X and Y be discrete random variables with joint probability mass function pX,Y (x, y) = C/[(x + y − 1)(x + y)(x + y + 1)], x, y = 1, 2, 3, . . . Determine the marginal mass functions of X and YProblem 1: Suppose X1,., X are multinomial counts, that is the vector (X1,..., X) follows a multinomial distribution Mult(n; p1,..., P) with joint mass function k k n! f(r1,., Tk) II |P, Pi = 1, r; = n. (0.1) j=1 j=1 (a) Consider the Dirichlet prior for (p1, ..., Pk): T(B1 + ... + Br) g(P1, .., Pk) = II B; > 0, EPj = 1, I(B1) x ... x I(B) j=1 j=1 where I(a) = wa-le-wdw is the gamma function. %3D Under this prior, derive the posterior distribution of (p1,., Pk). (b) Explain why the Dirichlet prior is a conjugate prior in this situation. (c) Note that the mean vector density in (1) is ( ) where a' denotes the transpose of a vector a. A common Bayes estimator of a parameter is its posterior mean. Give the Bayes estimator of an arbitrary p; in this situation (j e {1, ..., k}). (d) If k = 2, write the form of the Dirichlet prior. What is the familiar name of the Dirichlet distribution with k = 2 categories? |3D
- Previously, De Anza's statistics students estimated that the amount of change daytime statistics students carry is exponentially distributed with a mean of $0.88. Suppose that we randomly pick 25 daytime statistics students. Find the probability that an individual had between $0.80 and $1.00. Graph the situation, and shade in the area to be determined.Problem 2 Let(X, Y, Z) be a random vector with a joint pdf fx,x,z(x, y, z) · 6, 0 Z).Problem 2. Let Mx(t) = e + e + e. Find the following: (1) E(X). (2) V (X). (3) The probability mass function of X.
- HW 2 - Problem 2Problem 2.4: A joint discrete probability function f (x, y) is expressed in the table below. Find: (а) с (b) P(1 < X < 3, 1.5 < Y < 1.7) (c) P(2.5 < X < 3.5) (d) P((X +Y) < 4.6) (e) the marginal distributions fx(æ) and fy(y) (f) whether the random variables X and Y are statistically independent. f(x, y) 2.1 3.7 2.9 1.56 0.01 0.05 0.07 y| 1.64 0.04 1.72 0.21 0.17 0.02 0.03Problem 2: Find the moment generating function Mu(t) for the standard uniform random variable U (the continuous random variable whose density function is 1 on [0,1] and 0 elsewhere).
- Problem 7: Random variables X and Y have joint PDF 12e-(kr+4y), r 2 0, y 2 0, Sxx (2, y) = { 0, otherwise. (a) Find P[X +Y 2]. (c) Find P(max(X,Y) < 0.5]3. Let X₁, X2, Xn represent a random sample from a Rayleigh distribution with proba- bility density function given below. ...9 X f(x) = -=- e – x² / (20) Ө (a) Show that E(X²) = 20. Look up the gamma function, I(z) = f* 9 x > 0 t²-¹e-t dt. (b) Find E (Σ½-1 X²) and hence find an unbiased estimator of 0. i=1 (c) Estimate from the following observations using the estimator found in part (b). 16.88 10.23 4.59 6.66 13.68 14.23 19.87 9.40 6.51 10.95both questions please. thank you!