Let X₁, X₂ and X3 be a random sample of size n=3 from a distribution withp.m.f.Constructa)b)P(X= x) = ?3the probability mass function of; X= 1,the probability mass function ofX = X₁ + X₂ + X₂23S² = E; X= 2.Σ(X₁-X)²2

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Answered: Let X₁, X₂ and X3 be a random sample of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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 the joint probability distribution of X and Y isgiven byf(x, y) = 130 (x + y) for x = 0, 1, 2, 3; y = 0, 1, 2 construct a table showing the values of the joint distribu-tion function of the two random variables at the 12 points (0, 0),(0, 1), ... ,(3, 2).
1
9. 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: (f(x|y = 1)
Let X1,..., X10 be a random sample of size 10 from a N(u, o²) population. Suppose 10 Y =(X; - 4)? i=1 Find the probability that the random interval Y Y 20.5' 3.25 includes the point o?.
2. Suppose that X1, . Xn are iid Geometric random variables with frequency func- ... tion f(x; 0) = 0(1 – 0)", x = 0, 1, 2, ..., 0 E (0, 1). Find the ML estimator 0, of 0. Show that Ôn is consistent an find its asymptotic distribution.
a) Let X₁, X₂, X₁,...,X, be a random sample of size n from population X. Suppose that X-N(0, 1) and Y = -1-8√n. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal random variable. iv) What is the probability that Y2 is between 0.02 and 5.02?
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3. Let X₁, X, and X, be a random sample of size n=3 from a distribution with p.m.f. Construct a) b) P(X=x)= X ; x = 1, the probability mass function of X₁ + X₂ + X₂ 2 3 the probability mass function of ; x = 2. S² == Σ(x,-X)² 2

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Let X₁, X₂ and X3 be a random sample of size n=3 from a distribution with p.m.f. Construct a) b) P(X= x) = X= 3 the probability mass function of X₁ + X₂ + X₂3 3 the probability mass function of ; X= 1, S² = H ; X= 2. Σ(x,-X)² 2