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EBK AN INTRODUCTION TO MATHEMATICAL STA
- Consider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)arrow_forwardLet X and Y be discrete random variables with joint pdf f(x,y) given by the following table: y = 1 y = 2 y = 3 x = 1 0.1 0.2 0 x = 2 0 0.167 0.4 x = 3 0.067 0.022 0.033 Find the marginal pdf’s of X and Y. Are X and Y independent?arrow_forwardFor 50 randomly selected speed dates, attractiveness ratings by males of their female date partners (x) are recorded along with the attractiveness ratings by females of their male date partners (y); the ratings range from 1 to 10. The 50 paired ratings yield x=6.4, y=6.0, r=−0.170, P-value=0.238, and y=7.31−0.198x. Find the best predicted value of y (attractiveness rating by female of male) for a date in which the attractiveness rating by the male of the female is x=8. Use a 0.01 significance level. The best predicted value of y when x=8 isarrow_forward
- Let X1, ...., Xn be a random sample from a population with θ unknown and given by the density f(x; θ) = ( 1 2θ √2 x e − √2 x θ if x > 0 0 if x ≤ 0 1. Show that E(X) = 2θ 2 and E( √2 X) = θ (Hint: you may use that R ∞ 0 e −z z α−1dz = (α − 1)! for every α ∈ N). 2. Show that the statistic θbn := 1 n Xn i=1 p2 Xi (1) is an unbiased estimator of θ. 3. Give the definition of a consistent estimator. 4. Show that the estimator θbn given in relation (1) is a consistent estimator of θ. 5. Show that the estimator θbn is a minimum variance estimator of θ. (Hint: use the Cramer-Rao inequality given by var(θb) ≥ 1 nE ∂ ln(f(X;θ) ∂θ 2arrow_forwardFor 50 randomly selected speed dates, attractiveness ratings by males of their female date partners (x) are recorded along with the attractiveness ratings by females of their male date partners (y); the ratings range from 1 to 10. The 50 paired ratings yield x=6.3, y=6.0, r=−0.228, P-value=0.111, and y=7.81−0.280x. Find the best predicted value of y(attractiveness rating by female of male) for a date in which the attractiveness rating by the male of the female is x=5. Use a 0.10 significance level. The best predicted value of ywhen x=5 is nothing. (Round to one decimal place as needed.)arrow_forwardFind the maximum likelihood estimator for θ in the pdf f(y; θ) = 2y/(1 − θ^2), θ ≤ y ≤ 1.arrow_forward
- If X1, X2, ... , Xn constitute a random sample of size nfrom a geometric population, show that Y = X1 + X2 +···+ Xn is a sufficient estimator of the parameter θ.arrow_forwardConsider a random sample X1, … , Xn from the pdff (x; u) = .5(1 + (THETA)x) -1 <= x <= 1where -1 <= theta <= 1 (this distribution arises in particlephysics). Show that theta = 3X is an unbiased estimator oftheta. [Hint: First determine mu = E(X) = E(X).]arrow_forwardThe extent to which a distribution is peaked or flat, also called the kurtosis of the distribution, is often mea-sured by means of the quantity α4 = μ4σ4 Use the formula for μ4 obtained in Exercise 25 to findα4 for each of the following symmetrical distributions,of which the first is more peaked (narrow humped) thanthe second:(a) f(−3) = 0.06, f(−2) = 0.09, f(−1) = 0.10, f(0) =0.50, f(1) = 0.10, f(2) = 0.09, and f(3) = 0.06;(b) f(−3) = 0.04, f(−2) = 0.11, f(−1) = 0.20, f(0) =0.30, f(1) = 0.20, f(2) = 0.11, and f(3) = 0.04.arrow_forward
- For 50 randomly selected speed dates, attractiveness ratings by males of their female date partners (x) are recorded along with the attractiveness ratings by females of their male date partners (y); the ratings range from 1 to 10. The 50 paired ratings yield x=6.4, y=6.0, r=−0.143, P-value=0.322, and y=7.15−0.174x. Find the best-predicted value of y (attractiveness rating by female of male) for a date in which the attractiveness rating by the male of the female is x=7. Use a 0.05 significance level.arrow_forwardFor 50 randomly selected speed dates, attractiveness ratings by males of their female date partners (x) are recorded along with the attractiveness ratings by females of their male date partners (y); the ratings range from 1 to 10. The 50 paired ratings yield x=6.3, y=6.0, r=−0.264, P-value=0.063, and y=7.92−0.304x. Find the best predicted value of y (attractiveness rating by female of male) for a date in which the attractiveness rating by the male of the female is x=4. Use a 0.01 significance level. The best predicted value of y when x=4 is __ (Round to one decimal place as needed.)arrow_forwardLet X1...., Xn be a random sample of size n from an infinite population and assume X1 d= a + bU2 with the constants a > 0 and b > 0 unknown and U a standard uniform distributed random variable given by FU (x) := P(U ≤ x) = 0 if x ≤ 0 x if 0 < x < 1 1 if x ≥ 1 1. Compute the cdf of the random variable X1. 2. Compute E(X1) and V ar(X1). 3. Give the method of moments estimators of the unknown parameters a and b. Explain how you construct these estimators!arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage