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- Let X1,..., Xn be independent and identically distributed random variables with Var(X1) < ∞. Show that n 1 EjX;→p EX1. п(п + 1) j= Note. A simple way to solve a problem of showing Yn →p a is to establish limn EYn = a and limn Var(Yn) = 0.For E (0,1) let Xp be a Geometric random variable with parameter p. (a) Find a value of p so that P(Xp> 2.5) = 9. (b) Let An be the event that Xp is even. Determine P(A,) in terms of p. (c) Suppose Y, is a random variable which is equal to the remainder after integer division of X, by 3. Let p = i, and determine the conditional probability mass function of conditioned on the event Yı = 1.69. Let X~ Geom (p) with pmf f(x) = P(X = x) = p(1 − p)ª−¹I{1,2,...} (x). (a) Use the definition of the moment generating function of a discrete random variable to find the moment generating function mx (t) for X. (b) Use the mgf for X to derive the formula for the mean of X, μ = E(X) = m'x (0).
- B4. Let X₁,... Xn ~ N(μ, o2) be independent random variables. 2 (a) From lectures we know (X=X) ²³. X₂ (b) Let n i=1 ~ x²(v). What is the value of v? n 1 *³ =, ²-₁, [(x₁ - x) ². Σ(x s n 1 i=1 Determine Var(s), that is, the variance of the sample variance. (c) Assume now that we have observed data ₁,...,n ER with sample variance s². Use the result from part (a) to find numbers an and b, such that [ans, bns] is an exact 95%-confidence interval for o².11. Let (y1, Y2 ... Yn) be independent random sample from the uniform distribution on [0, 1]. (a) show that Z = – In Y; has exponential distribution with parameter 1. (b) Hence or otherwise, show that -2 In Y; xảnSuppose that the moment-generating function of a random variable X is given by Mx(1) = +" +" + 4 3t 2 4 5t 15 Find the probability mass function of X.
- Let X be a discrete random variable with the following PMF 1 1 for x=-2 8 100 for x = -1 Px(x) = for x = 0 1 for x = 1 for x = 2 4 0 otherwise a) Find E[X] and Var(X) b) Now we define a new random variableY = (x+1)², Find PMF of Y and E[Y]5. Let X₁ and X₂ be iid Poisson() random variables with pmf p(x) = H'e exp{p(e-1)), for tER. Let Y=X₁+X₂. a. b. What is the distribution of Y? What are E[Y] and Var[Y]? for x 0,1,2,..., and mgf 9 X!2. (a) Suppose X,~ N(0,6), for i = 1, 2, 3, 4. Assume all these random variables are independent. Derive the value of k in each of the following. i. P(X₁ + 8X₂ (k(X² + X² + X²))¹/²) = = 0.05.
- Suppose that X₁, X₁, X3 are mutually independent random variables with the respective moment 2 generating functions, Mx, (t) = e²¹², Mx₂ (t) = e²t + 3t², and Mx₂(t) = (¹₂) ². 1-2t, Find the value of such that P(Y > t) = 0.005, for Y = 21Let X₁,..., Xn be a random sample from a geometric distribution, X~ GEO(p). Here, -1 P[X = x] = p(1 − p)*−¹ for x = The method of moments unbiased estimator for Var [X] None of the other answers ·Σ" (X; – X)² i= X² n 1 [X² -x] = 1, 2,... n+1 = 1-p p² isSuppose that the random variables Y1,..., Y(n > 2) satisfy Y - Ba, + Ei, i= 1,2,..., T, where 1,..., En are iid N(0, o), and both B and o are unknown. (a) Assume z1,..., In are fixed known constants. Here we observe Y1 = n,., Y observed data Y = y = (y1,..., Yn). Find a two-dimensional sufficient statistic of Y = (Yı,., Yn) for (8,0?). (b) Assume now that a1,...,n are random variables with a known joint distribution m(r1,..., In), and the r,'s are independent of G's (it is traditional in the linear regression to use lower case for independent variables r;'s). In this case, the observed data (Y, x) = {(Yi, r;)}i-1,n. Find a three- dimensional sufficient statistic of (Y, x) for (B,02). Im, and the