1. Fix r € N, and suppose that Y₁, ..., Yn are independent and identically distributedNegBin(r, ) random variables, each with probability mass function given by¡v (u | 0) – (1 + r − ¹) (1 -fyy(1 - 0) ³0⁰for y ¤ N and unknown 0 € (0, 1). Here the number of successes r in the negativebinomial is known. (a) By specifying appropriate functions h(y), U(0), T(y) and A(U(0)), show thatthis parametric family is an Exponential Family.(b) Write down a minimal sufficient statistic for 0, justify your answer.dA(c) Using that E{T(Y)}for the parameter==(or otherwise), write down an unbiased estimator

Given that are independent and identically distributed random variables, each with probability…

Answered: 1. Fix r € N, and suppose that Y₁,...,…

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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2. a. If X is a discrete random variable with probability mass function P(X =x) and a, b are constants, prove that E(aX +b)= aE(X)+b, ii. Var (ax +b)=a³Var(X). i.
For E (0,1) let Xp be a Geometric random variable with parameter p. (a) Find a value of p so that P(Xp> 2.5) = 9. (b) Let An be the event that Xp is even. Determine P(A,) in terms of p. (c) Suppose Y, is a random variable which is equal to the remainder after integer division of X, by 3. Let p = i, and determine the conditional probability mass function of conditioned on the event Yı = 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)
The annual rainfall (in inches) in a certain region is normally distributed with µ = 40 and σ = 4. What is the probability that starting with this year, it will take more than 10 years before a year occurs having a rainfall of more than 50 inches? Suppose the rainfall in each year is independent of the rainfall in other years and the distribution of rainfall in each year is the same as in the present year.
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.
Suppose that the moment-generating function of a random variable X is given by Mx(1) = +" +" + 4 3t 2 4 5t 15 Find the probability mass function of X.
13. Suppose that the joint probability density of two random variables X₁ and X₂ is given by: J6e-2x₁-3x2 for x₁ > 0 and x₂ > 0, otherwise 0 f(X1.X2) = {6 Find the following probabilities. Show all steps. a) P(1 ≤X₁ ≤ 2, 2 ≤X₂ ≤3) b) P(X₁ 2).
7. 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: The marginal function of x %3D
Q 6.1. Suppose Z = (Z₁, Z2, Z3) is a standard multi-variate Gaussian random variable i.e., for i ≤ 3, Zi~ N(0, 1) are i.i.d. random variables. Each of the random variables, (a)–(d), on the left is equal in distribution to exactly one random variables, (1)–(4), on the right. Pair up according to "equal in distribution" and explain briefly your reasoning. X₁ (a) X₂ (b) (x₂) - (2) (c) (x²) - (¹/1² 1/√²) (2₁) (3/√2 = √2 = (V2) Z₁ (d) (x) - (1) (²) X2 Y₁ (9)-(-) (2) (1=1) Y₂ (1) Y₁ (²) (4) - (1/² (3) (1/√√2 1/√2-1/√2) (x₁) = (²₁ ² ✓/³²) (3) 2 2 √2 1 Z3 Y₁ → ()-()) (4) = Y₂ 1/2) (2) 1) (2)
Show that two random variables X, and X, with joint pdf, 1 6. 1 |x, |< 4, 2

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