Let the combined density function of the two -dimensional continuous random variable (x, y) are p (x, y) = {1, | y |
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1. Let the combined density function of the two -dimensional continuous random variable (x, y) are p (x, y) = {1, | y | <x, 0 <x 10, other condition density
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- Let X₁, X₂,..., X, be independent, uniformly distributed random variables on the interval [0, b]. a) Find the probability distribution function of X(n) = max(X₁, X₂,..., Xn). Fx (t) = for 0 ≤t≤ b and zero elsewhere I b) Find the probability density function of X(n)- fx (t) = c) Find the expected value of X(n) E(X(n)) = for 0 ≤t≤ b and zero elsewhereQ1) CDF, PDF, Expectation and Variance The cumulative distribution function (CDF) of random variable Y is y 1. a) Find the probability density function, fy(y), of random variable Y. b) Show that f fy (y)dy = 1. c) What is E[Y]? d) What is VAR[Y]?A bivariate random variable (X, Y) has joint probability density function (pdf) { 8xy, 0 |X = })? (c) Let the random variables U and V be defined by X U = Y' V = Y. (i) Find the joint pdf of (U, V) and clearly specify the domain of U and V. (ii) Find the marginal pdf of U.
- 4. Let X and Y be independent, continuous random variables with densities and fx(x) = = fy (y) = = 0 {} if 0 < x < 2 otherwise. y if 0Let (X; Y ) be a continuous random vector with joint probability density function 0.5 -1Let X1, X2,...,X, be a random sample from a distribution with density function e if x > 0 f(x; 0) elsewhere What is the maximum likelihood estimator of 0 ?Let X and Y have the joint probability density functionf(x, y) = 5, for 0 < x < y < 1.(i) Find the marginal probability density function of X and Y.(ii) Find the conditional probability density function of X, given Y = y.(iii) Find the conditional mean of X, given Y = y.(iv) Find the conditional variance of X, given Y = y.4) The joint density function of the random variables X and Y is given by (&xy f(x, y) = 0SXS1,0 sysx otherwise Find (a) the marginal density of X, (b) the marginal density of Y, (c) the conditional density of X, (d) the conditional density of Y.6. Let X and Y be continuous random variables with joint density function 24xy if 0 < x < 1,0 < y < 1 – x f (x, y) = 0. otherwise. Calculate E(Y|X = }).Let Y₁, Y₂,..., Yn be a random sample from the inverse Gaussian distribution with probability density function: f(y, μ, 2) = { Where μ and are unknown. a) What are i. 1 -ACT √ λ 2ny3)že elsewhere if y > 0 the likelihood ii. the log likelihood functions of u and λ. b) Find the maximum likelihood estimators of u and 2.Asapb) Let Y,,Y2, .. , Yn denote a random sample from N(0,0) distribution with probability density function: f(y;8) = e V2n0 i) Show that f(y; 0) belongs to the 1-parameter exponential family. ii) What is the complete sufficient statistic for 0? Justify your answer. iii) Show whether or not, the maximum likelihood estimator is an unbiased estimator of 0. iv) Does the estimator attains the minimum variance unbiased estimator of 0.Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON