8.18. Let X be a binomial random variable with parameters n and p, which meansthat its mass function is given byƒ(1) = (*)p'′(1—p)*-', i=0,...,n,and 0 otherwise. Prove that810=(1+x)= 1-(1-p)"+1(n+1)p

X~B(n,p)Probability mass functionf(i)=n i pi(1-p)n-i. i =0,1,2,...n Formula Used for expected…

Answered: 8.18. Let X be a binomial random…

A First Course in Probability (10th Edition)

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Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Suppose that Y and Z are random variables with Y~ Bernoulli(p) and Z = + 1. Which of the following gives the joint probability mass function P(Y = y, Z = z) for y = 0, 1 and z = 0, 1? Select one: O a. None of the other choices — O b. (1-y)zp+ y(1 − z)(1 − p) - - O c. (1-y)(1 z)p+yz(1 − p) ○ d. yzp+ (1 −y)(1 — z)(1 − p) - O e. y(1-z)p+ (1 − y)z(1 − p) -
of 9 A discrete random variable X has probability mass function Sk() for a = 0, 1, .. . , 16 px(x) = otherwise, where k is a constant that makes px a valid probability mass function. Calculate P(X = 8). Note: You first have to calculate the value of k that makes px a valid probability mass function.
8.18. Let X be a binomial random variable with parameters n and p, which means that its mass function is given by ƒ(1 (*)p²(1–p)**, = (*) p² (1-p)*-*, i=0,...,, and 0 otherwise. Prove that E 1+X = 1-(1-p)"+1 (n+1)p
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B4. Let X₁,... Xn ~ N(μ, o2) be independent random variables. 2 (a) From lectures we know (X=X) ²³. X₂ (b) Let n i=1 ~ x²(v). What is the value of v? n 1 *³ =, ²-₁, [(x₁ - x) ². Σ(x s n 1 i=1 Determine Var(s), that is, the variance of the sample variance. (c) Assume now that we have observed data ₁,...,n ER with sample variance s². Use the result from part (a) to find numbers an and b, such that [ans, bns] is an exact 95%-confidence interval for o².
5. Let X₁ and X₂ be iid Poisson() random variables with pmf p(x) = H'e exp{p(e-1)), for tER. Let Y=X₁+X₂. a. b. What is the distribution of Y? What are E[Y] and Var[Y]? for x 0,1,2,..., and mgf 9 X!
Suppose that the random variables Y1,..., Y(n > 2) satisfy Y - Ba, + Ei, i= 1,2,..., T, where 1,..., En are iid N(0, o), and both B and o are unknown. (a) Assume z1,..., In are fixed known constants. Here we observe Y1 = n,., Y observed data Y = y = (y1,..., Yn). Find a two-dimensional sufficient statistic of Y = (Yı,., Yn) for (8,0?). (b) Assume now that a1,...,n are random variables with a known joint distribution m(r1,..., In), and the r,'s are independent of G's (it is traditional in the linear regression to use lower case for independent variables r;'s). In this case, the observed data (Y, x) = {(Yi, r;)}i-1,n. Find a three- dimensional sufficient statistic of (Y, x) for (B,02). Im, and the

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8.18. Let X be a binomial random variable with parameters n and p, which means that its mass function is given by ƒ(1)= (*)p'(1-p)*-', i=0,...,”7, and 0 otherwise. Prove that 810 = [1 + x) = 1 = (1-P) ²+1 (n+1)p

8.18. Let X be a binomial random variable with parameters n and p, which means that its mass function is given by ƒ(1)= ()p'(1-p)-', i=0,...,”7, and 0 otherwise. Prove that 810 = [1 + x) = 1 = (1-P) ²+1 (n+1)p