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- We 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 JO, J1,..., J4 independent random variables according to the Ber (r;) law, where i = 0, 1,..., 4, respectively. We define the random variables Xi = min {JO + Ji, 1}, for i = 1, 2, 3, (a) Find the law of Xi , for each i = 1, 2, 3, 4. (b) Find the law of (X1, X2, X3, X4).11. Assume that X is a uniform random variable on the interval [-22, 14. (a) tion of its mean. In other words, compute Find the probability that X is within two standard devia- P(µ – 20 < X < µ +2o). -
- 2. Suppose that X1, . Xn are iid Geometric random variables with frequency func- ... tion f(x; 0) = 0(1 – 0)", x = 0, 1, 2, ..., 0 E (0, 1). Find the ML estimator 0, of 0. Show that Ôn is consistent an find its asymptotic distribution.Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p 0, with probability 1-p} for every i = 1, 2, ... , n. Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?9. If X and Y are two random variables and let g(X) be a random variable. Show that (a) E[g(X) X=x] = g(x). (b) E[g(x)Y|X=x] = g(x) E[Y|X=x]. Assume that E[g(x)] and E[Y] exist.
- 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 o².2. Let the independent random variables X1 and X2 have Bin(0.1,2) and Bin(0.5, 3), respectively. (a) Find P(X1 = 2 and X2 = 2). (b) Find P(X1 + X2 = 1). (c) Find E(X1 + X2). (d) Find Var(X1 + X2).Let X = b(n, and E(3x) = E(3-5x) find n.
- Theorem 11. Let X be a random variable and let g(x) be a non-negative function. Then for r > 0, Eg (X) P[g(X) > r] < Proof.Suppose Y is a continuous random variable drawn from the uniform distributionon the interval [3, 4], that is, Y ∼ Uniform([3, 4]). Conditioned on Y = y, a second randomvariable X is drawn from the uniform distribution on the interval [0, y]. What is fX(x), thepdf of X?4. Assume that X is an exponential random variable. Suppose further that Var(X) = 5. E(X). (a) Find the parameter 0> 0. (b) Compute Var(X + 4). (c) Compute P(X > 15[X > 11).
