1. Suppose that the random variables X and Y have the following joint pdf: f(x,y) = { }(2x+3y) for 0≤a ≤ 1, 0≤ y ≤ 1, otherwise. (a) Find the marginal densities, fx (x) and fy (y). (Be sure to give the support set for each random variable.) (b) Compute P(X < 0.6). (c) Compute P(Y ≥ 0.5). (d) Find E(X). (e) Find E(Y).
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- b) Let X₁, X2, X3,...,Xn be a random sample of n from population X distributed with the following probability density function: f(x;0)=√√2n0 0, -20₁ if -∞0 < x <∞0 otherwise (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. (iii) Check whether or not the estimator obtained in (ii) is unbiased. (iv) Find the Fisher information in this sample of size n about the parameter 0.2. Assume X ~ Exp(0), so that its pdf is 1 f(x; 8) x > 0,0 < 0 < o. e We have 5 samples, 1,2,3,4, and, 5, under Exp(0). Find the maximum likelihood estimator of 0 using the five samples.We have a random variable X and Y that jave the joint pdf f(x) = {1 0<x<1, 0<y<1} {0 otherwise} Let U = Y - X2. What is the support for the random variable U? Are there critical points? If U = Y/X. What is the support for the random variable U? Are there critical points?
- 3. Let the random variable X have the moment generating function M(t) = What are the mean and the variance of X, respectively? -1 < t < 1.7. Given that f(x, y) = (2x+2y)/2k if x = 0,1 and y = 1,4, is a joint probability distribution function for the random variables X and Y. Find: The marginal function of x %3DSuppose W is a random variable with the pdf f(w) = 3u3e-w³/343 for w≥ 0.1 If Y = -12W + 11, what is the pdf of Y? е
- 2) Let X₁, X2, ..., Xn be a random sample from the pdf f(x;0) = 0xª−¹, 0≤x≤ 1,0 <0 < ∞. Find the MLE of 0 and show that its variance approaches 0 as n approaches ∞o.Let Y < Y, < Y3 be the order statistic of a random sample of size 3 from the uniform distribution having pdff (x;0) = 1/0,0Given that f(x, y) = (2x+2y)/2k if x = 0,1 and y = 1,4, is a joint probability distribution function for the random variables X and Y. Find: The marginal function of X ,f1(X)Let X, and X; be independent exponential random variables having pdfs f, (x) = e, x > 0 and fr,(x2) = e"", I3 > 0. Given that Y, = X,/X and Y, = X, + X2. a. Find the joint pdf of Y, and Y2 b. Find the marginal pdf of Y1.12. Let the random variable X and Y have joint pdf 4 f(x,y) = (x² + 3y²), 04. 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).Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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