**Problem #4**Let the random variable \( X \) have a Poisson distribution with parameter \( \lambda \). Use Chebyshev's inequality to show that \( P(X \geq 2\lambda) \leq \frac{1}{\lambda} \).

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Answered: Problem # 4 Let the random variable X…

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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11. Let (yı, Y2, .., Yn) be independent random sample from the uniform distribution on [0, 1]. (a) show that Z = – In Y; has exponential distribution with parameter 1. (b) Hence or otherwise, show that –2) In Y; Xản i=1
(a) Let X, Y be independent random variables with respective PMFS Px(x) = (-) )* > Py (Y) = (-) )", y=1, 2, 3, . . . .2 , х-0, 1, 2, 3, . . Find P(X+Y=3)
Q.7) Suppose X is a random variable with probability mass function given in Question 5 (**see below) a.) Find E(X4| -1< or equal to X < or equal to 1) **Here's Question #5 Suppose X is a random variable with probability mass function given in the following table: x = -2.0, -1.0, 0.0, 1.0, 2.0 p(x) = 0.4, 0.21, 0.0, 0.13, 0.26
A continuous random variable X has the distribution function if X 3 Find K and f (x), P.d.f.
2) Let X₁, X2, X3, X4 be a random sample of size 4 from a population with the following distribution function Where, ß > 0. If fram e ¹- {** f(x;0)=B for x > 4 otherwise "1 g) What is the maximum likelihood estimator of g (B) = 2(ß + 1). h) Is the maximum likelihood estimator of g (B) = 2(B + 1) unbiased or not?
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Example 4-28. (Buffon's Needle Problem). A vertical board is ruled with horizontal parallel lines at constant distance 'a' apart. A needle of length l (< a) is thrown at random on the table. Find the probability that it will intersect one of the lines.
Question 3. Let X₁,, Xn be a random sample from a distribution with the pdf given by f₁₁x(x) = exp(-ª) if x ≥ A, otherwise f(x) = 0, where > 0. Find the MLE's of 0 and X. Start by writing the likelihood function and note the constraint involving A. Question 4. #4.2.9 of the textbook.
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11. Let (y1, Y2 ... Yn) be independent random sample from the uniform distribution on [0, 1]. (a) show that Z = – In Y; has exponential distribution with parameter 1. (b) Hence or otherwise, show that -2 In Y; xản
Suppose there are three possible states of nature, and the class-conditional PDFS are the Cauchy distributions 1 p(rw;) = 1 i = 1,2,3. (1) 2 1+ bị Let ai = -2, a2 = 0, az = 2, bị = b2 = 1, b3 = 0.5. Let P(wi) = P(w2) = 0.4, P(w3) = 0.2. (a) Find the decision boundaries and the decision regions for the Bayesian decision (b) Calculate the probability of error for the classification done according to this
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Problem # 4 Let the random variable X has a Poisson distribution with parameter ) Usa Chehyebev's inequality to show that P( Y 2)) 1/)