If X₁, X₂, X3, .., Xn are random sample from a population with mean μ and variance o², then what is ε[(x₁ - μ)(x₁ - μ)] for ij, i = 1,2,3,.... ..., n?
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![If X₁, X₁, X3, ..., xn are random sample from a population with mean µ and variance o², then what is
ε[(x₁ - μ)(X; -μ)]
for ij, i = 1,2,3,..., n?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbba17155-8caf-49af-9c4c-39c2bffb82ec%2F37150d83-0d49-49e4-b256-87c0bd85aa6d%2Faqvwi3q_processed.png&w=3840&q=75)
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- 1. Let X1, X,.., X, random sample from distribution be a a Geometric 2....., fx (x) = p(1– p)*| F, (y,) = ? x-1 x=1,2,3,.... Fy, (y,) = ? P(Y, <1) =? а. b. С.If we let X1, X2,...X10 be a random sample from the distribution N( μ, σ), then what is the value k so that: P( (X-μ)/(σ/√10) < k ) = .05 P( (X-μ)/(S/√10) < k ) = .05 P( k < (X-μ)/(S/√10) ) = .05Suppose X,,X,,...,X, is a random samples from normal distribution with mean, µand variance, n ΣΧ Let Y,, Y2,., Y, be an independent chi square distribution, each with 1 i=1 1 and X= degree of freedom. State the distribution and parameter(s) of each of the following random variables. a) J= \n(X – u)
- 16. Let X, X2 X, be a random sample from a normal distribution, X, N(6, 25), and denote by X and S2 the sample mean and sample variance. Use tables from Appendix C to find each of the following: (a) P[3 < X < T. (b) P[L860< 3(X-6)/S]. (c) PIS <319375]25. Suppose X is a random variable distributed with mean u and standard deviation a > 0. If the random variable Y is defined as Y = u(X - uX)/a, then E(Y) is: a. 0 b. 1 d. E(X)3. If x1, X2, X3, ..., Xn is a random sample from a normal 1 distribution N (µ, 1) show that t= E xí is an unbiased estimator of n i = 1 H² + 1.
- X, X, and X, is a random sample of size 3 from a population 2. with mean value u and variance o?, T., T, T, are the 2' *3 estimators used to estimate mean value u, where T=X,+X,-X, T,= 2X,+3X, - 4X, & T, %3D 3 2 1. 3. 2. 3 1 =(ax,+X,+X,) 3 1. (i) Are T, and T, unbiased estimators ? 1 2 (ii) Find the value of A such that T, is unbiased estimator 3 for u.The random variable Z has mean 30 and variance 225. Calculate the limits between which you would expect the bulk of observations to lie for the random variable Z/3 − 7.Let X1, …, Xn be a random sample from a N(μ, 1) population. Find the MLE of μ.
- The amount of gasoline in gallons sold by three different gas stations during one day is given by the independent random variables X1,X2, X3 each with a normal distribution. X1 has a mean µl =700 and standard deviation ơ1 55; X2 has mean u2 =700 and standard deviation o2 = 65; X3 has mean u3=900 and standard deviation 03 = 100. Suppose the prices per gallon are $2.90, $3.00 and $3.10 for X1, X2, and X3 respectively. Find the probability that the combined revenue for a given day is less than $6000. Use the z-score table. Round answer to the nearest hundredth.12. Let (*1, 12, ., r„) be independent samples from the uniform distribution on [0,0]. Let X(n) and X(1) be the maximum and minimum order statistics respectively, (a) Find the distribution of X(m) and X(a) and hence, their means and variances. X{n) (b) Show that 2nY x where Y = – In %3D (c) Show that 2) In - Xản: Hence write a function of the geometric mean, i=1 (II GM(r) = which is Xn 19. Let X₁, X2, ..., X5 be a random sample from a distribution with mean and variance 5 X₁, find Cov(X₁, X) 2. If X= = i=1
