Let X1,...,Xn be a random sample from a distribution with mean μ and variance σ2. Find the value of the following quantities (as a function of μ and σ2). (a) E(X1 −X2 +X3 −X4) (b) Var(X1 − X2 + X3 − X4) (c) E(4X1 −3X2 +2X3 −X4)
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Let X1,...,Xn be a random sample from a distribution with
mean μ and variance σ2. Find the value of the following quantities (as afunction of μ and σ2).
(a) E(X1 −X2 +X3 −X4)
(b) Var(X1 − X2 + X3 − X4)
(c) E(4X1 −3X2 +2X3 −X4)
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- and I respectively. Their correlation coefficient is 0.4. New random variables W and V are defined as Two random variables X and Y have meáns-í and 2 respectively and variances V = - X + 2Y; WX + 3Y Find the (c) correlations (d) correlation coefficient of V and WSuppose you have a joint distribution of x and y where • x has population mean ug and population variance o? • y has population mean ly and population variance o, and the population covariance between them is Ery. The population Pearson correlation is then given by Ery OxJy We collect n pairs of data, (X;, Yi), i = 1,.. , n. Each pair is an independent draw from this distribution. (a) Suppose we estimate our population covariance with n 1 (x; – Ha)(yi – µy). i=1 Is this an unbiased estimator of the population covariance? Show why or why not. (b) Suppose we estimate our correlation with (xi – Ha)(Yi – Hy) - n i=1 Ox0y Is this an unbiased estimator of the population correlation? Show why or why not. =WI7. Let X,, X, X, and X, be pairwise uncorrelated random variables each with zero mean and unit variance. Compute the correlation between (a) X +Xz and X2 + X3. (b) X +X, and X3 + X4. (c) X-X2 and X2- X3. (d) X,-X2 and X3 - X4.
- Q1. Let X; and S be the mean and variance of independent random samples of size ni from populations with mean H; and variance of (i = 1,2), respectively where u, = 20. of = 9, n1 = 36 and u2 = 18, ož = 16, n2 = 49. Find (- a) P(X, > 20.98) b) x such that P(X, 12.8062) d) si such that P(S} > s3) = 0.80 e) P(X1 – X2 > 0.2336) n P(SE/S글 > 0.8368) %3D3) Suppose X is a discrete variable that has the following pr function (pdf) f (X) 1. 0.40 0.20 3 0.15 4. 0.25 a) Calculate the cumulative distribution function (the b) Find the expected value, showing your work: E(X) c) Find the variance, showing your work: Var (X)[Q1] The probability distribution of the discrete random variable X is: f(X) = k23-* , for x = 4,5, 6, 7 (a) Find: • the value of the constant k. • The mean E (X). • The variance Var(X). (b) Sketch F(X).
- 6. Show that 1 s2 E1(Xi – x)² is unbiased estimator of the population variance o? i=1 п-1Q1. let Y₁ < Y₂ < Y3 < Y4 < Y5 are the order statistics of the random sample of size 5 from the distribution : f(x) = 3x², 02. The random variable X is the amount of fructose in apples (in ounces) grown at a local orchard and is represented by the following cdf: 2(x +), 1Let the random variables X and Y have the joint p.m.f. x+ y f (x, y) = 32 x = 1, 2, y = 1,2,3,4. Find the means lx and uy, the variances oy and oý, and the correlation coefficient p. Are X and Y independent or dependent? 6.1. A discrete random variable X follows the Uniform distribution if X takes values x = 1, 2, .., N, with P(X = x) = 1/N. Compute E(X), E(X²) and the variance Var(X). You may use the following identities: п(п + 1) (1) 2 п(п + 1)(2n + 1) (2) i=1(a) Let Y be a random variable distributed as X. Determine E(Y) in terms of r. (b) Let {X1, X2, . .. , Xn} be a random sample drawn from a normal distirbution with mean u and 1 variance o?. Denote S E-(X; – X)² as the sample standard deviation. Use the 1 n - result in part (a), or otherwise, to find E(S). (c) Find an unbiased estimator for the population standard deviation o.SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON