Assume a series of n independent and identically distributed Bernoulli randomi.i.d.variables: B;Bernoulli(p) for i = 1, 2, 3, ..., n.Question part 2:Derive the MGF of X = B1 + B2+...+Bn = D"1 B¡Hint: recall that for independent random variables X and Y and functions f() and g(),EEO n(1 – p+ pe*)enptо (1-р+pе')"O n(e*(1 – p) + p) Assume a series of n independent and identically distributed Bernoulli randomi.i.d.variables: B;Bernoulli(p) for i = 1, 2, 3, ..., n.Question part 1:Derive the mgf of any one of the B; (since they are i.i.d., each one has the same mgf)et +101-p+ pe*O e' (1 – p) + pO etp

Answered: Image /qna-images/answer/5892d196-f719-464f-a851-2dcf5e06728d.jpg

Answered: Assume a series of n independent and…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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1. The random variables §, §1, §2,... are independent and identically distributed with 1/4 and P(§ = j) = c/j for j = 1,2,3. Let X₁ = 0 and Xn = max(§₁, ..., Èn) for n = 1, 2,.... distribution P(§ = 0) = (a) What value must c take? (b) Explain why {Xn, n = 0, 1, 2,...} is a Markov chain. (c) Write down the transition matrix. (d) Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc). (e) Calculate P(Xn = 0). (f) Calculate P(X₁ = 3, X₂ = 1|X₁ = 3).
Q7. Let X; (i = 1,.,n) be a random sample from the N(u, o²) population with unknown parameters u and o?. Then we know that T, = E-1(X - X)²/(n – 1) is an unbiased estimator -xi-1 . Now consider two other estimators T, and T3 for o? where of a? and further, (n-1)T T2 = E(X – X)*/n and T3 = E(X – X)²/n + 1). %3D What is the variance of T,? Find out the variances and biases of T2 and T3 and thus obtain their m.s.e.'s. Show that among the three estimators T,, T2 and T3 the one with minimum m.s.e. is T3.
3. Let X1, .… , Xn be a random sample. Find the mle of 0 for the cases that the population pdf/pmf is, f(r;:0) = 0, %3D otherwise
Please answers the last three questions The random variables ξ, ξ1, ξ2, . . . are independent and identically distributed with distribution P (ξ = 0) = 1/4 and P (ξ = j) = c/j for j = 1, 2, Let X0 = 0 and Xn = max(ξ1, . . . , ξn) for n = 1, 2, . . .. What value must c take? Explain why {Xn, n = 0, 1, 2,..... } is a Markov Write down the transition Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc). Calculate P (Xn = 0). Calculate P (X4 = 3, X2 = 1|X1 = 3).
Let X; (i = 1, 2,..., n) be i.i.d. exponential random variables with E{X;} = 1/A. Let Y = mn{X1, X2,..., Xn}. What is the distribution of Y? (Compute P{Y > y}.) What is the distribution of Z = max{X1, X2, ..., Xn}? (Compute P{Z < z}.)
Let Y be a random variable with pdf f(y)=(3/64)y2(4-y), Osys4, zero elsewhere. Match the following. A. 0.31 B. 12.2 C. 16 D. 2.40 E. 0.64 select 1. E(Y)= select 2. V(Y)= select 3. Let X=3Y+5= E(X)= select 4. Let Z-5Y+3= V(Z)= select 5. P(YS2)3
Let Z1 and Z2 be independent standard normal N(0, 1) random variables. Let X = Z1 + 2Z2, and let Y = 2Z1 + Z2. a. Find rho( X, Y ) = correlation of X and Y. b. Find the numerical value of P{ X > 0 and Y > 0 }. HINT: Use trigonometry.
Number 2
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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