Define a random process X(t) as follows: (1) X(t) assumes only one of twopossible levels 1 or -1 at any time, (2) X(t) switches back and forth betweenits two levels randomly with time, (3) the number of level transitions in anytime interval t is a Poisson random variable, that is, the probability of exactlyk transitions, when the average rate of transitions is 1, is given by[(t)/k!] exp(-), (4) transitions occurring in any time interval are sta-tistically independent of transitions in any other interval, and (5) the levelsat the start of any interval are equally probable. X(t) is usually called therandom telegraph process. It is an example of a discrete random process.(b) Find probabilities P{X(t) = 1} and P{X(t) = -1} for any t.

Given the description, the process X(t) is a random telegraph process. Since it only has two…

Answered: Define a random process X(t) as…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

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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A lot of 25 light bulbs consists of N₁ defectives and N₂ working bulbs. The buyer accepts the lot of N₁ + N₂ = 25 bulbs if the number of defectives, X among n = 5 items taken at random and without replacement from the lot is less than or equal to 1. The operating characteristic curve is defined as the probability of accepting the lot, namely OC (p) = P(X ≤ 1) = P(X = 0) + P(X = 1), where p = N₁/25 is the probability of a bulb being defective. (a) Show that the probability of the picking exactly X defectives follows a hypergeometric distribution. (b) Determine the operating characteristic curve for p = 0.04, p = 0.08, p = 0.12 and p = 0.16. (c) Plot, using R, the operating characteristic curve for N₁ = 1 to N₁ = 4.
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Which of the following statements are true (no need to justify)?(1) The transformation of a continuous random variable is always a continuous random variable.(2) The distribution of the transformation of a random variable is always the same as that of the original sat-distribution of the univariate.(3) Many random variables can be represented by a uniformly distributed random variableusing if the quantile function of the distribution is known.(4) The accumulation function of several variables is increasing with respect to each variablefunction.(5) The multidimensional accumulation function is determined by its (one-dimensional) marginal distributionwhere from.(6) The marginal distributions of a discrete random vector are discrete distributions.

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