Exercise 6. Suppose a random variable has support: {0, 1, 2,4} and its pmf is:X0124f(x) 3/7 2/7 1/7 1/7(a) Calculate an explicit expression for the moment generating function: M(t).(b) Using your moment generating function, calculate E(X²) and E(X), and thereby the variance0².

Let X be a random variable with probability function (probability mass function or probability…

Answered: Exercise 6. Suppose a random variable…

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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4b)
Consider an estimator of μ: W= 1/6 Y1+1/16Y2+1/4Y3+1/8Y4+1/2Y5 This is an example of a weighted average of the Yi's. Show that W is also an unbiased estimator of μ. Find the variance of W.
(16) The moment-generating function of the geometric random variable X with parameter p is M(t) = 1-per. Use this to find the mean and variance of X.
Suppose that the response y is generated by y = f(x) + €, where e is a zero-mean Gaussian noise with variance 1. a) Suppose that f(x) = x. Randomly generate 10 x's and generate the corresponding y's; you need to generate two random numbers (i.e., x and e for each of the 10 points). Fit the data with linear regression and plot the scatter points. b) Suppose that f(x) = x². Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points. c) Suppose that f(x) = 1/x. Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points.
Let X1 and X2be independent exponential random variables: fX1(x1) = e−x1 and fX2(x2) = e−x2 1
f (x) = 4xc for 0 ≤ x ≤ 1 ; Find the constant c so that f (x) is apdf of some random variable X, and then find the cdf, F (x) = P(X ≤ x). Sketch graphsof the pdf f (x) and the cdf F (x), and find the mean μ and variance σ2.
Consider the one-way analysis of variance model Xij = µ + a; + Eij, i= 1,.., m, j= 1,..., ni, where ɛij ~ N(0,0²) are independent. Let n= n1 + · ·+ nm, ... ni 1 ni 1 X;. >Xii for i = 1,..., m, and X. = - ΣΣΧ. ni j=1 i=1 j=1 (a) Show that SS(TO) = SS(T) + SS(E), where SS(TO) = E E i=12j=1 (Xij – X..)², m m ni SS(T) Σι (X.-Χ.) and SS(E) -ΣΣ (Χ- X.) . i=1 i=1 j=1
Let G and H be two independent unbiased estimators of 0. Assume that the variance of G is two times the variance of H. Find the constants a andb so that aG + bH is an unbiased estimator with the smallest possible variance for such a linear combination.
Suppose that y, is an independent response variable (i = 1.....n) with mean, such that g(i) = n₁ = X₁B, where g(.) is a link function, X, is a vector of covariates and 3 is a px 1 vector of coefficients. Let the variance of y; be Var (Y) = V(i), where V(.) is a known function and is a scale parameter. Given: Zi= g'(pi) (Yi - Pi) + Ni w₁ = [V (pi) (g'(pi))²]-¹ 8 where g'(μ) = 9(μ) (a) Show that E(Z₁) = X₂B (b) Show that Var (Z₁) = w¹o. (c) Estimate 3 by minimizing Σ, wi( E(Z₁))² with respect to 3. After you found 8, find the Var(8) and E(3). What do you conclude?

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Exercise 6. Suppose a random variable has support: {0, 1, 2,4} and its pmf is: X 0 1 2 4 f(x) 3/7 2/7 1/7 1/7 (a) Calculate an explicit expression for the moment generating function: M(t). (b) Using your moment generating function, calculate E(X²) and E(X), and thereby the variance 0².