Define a random process ✗(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 λ, is given by[(t)/k!] exp (−t), (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.(a) Find the autocorrelation function of the process.

Given that a random telegraph process with the following properties:1) assumes only one of two…

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

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 32E

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II. Let X be a discrete random variable with possible values xi, i=1,2,.., and p(xi)=P(X=xi) is called the probability distribution function of X. We have > p(x,)=. The expected value, also called expectation, average, or mean, of X is defined as u= E(X)=, For any function g(x), x=1,2,3,.., E(g(x)) =. . The variance of X: Var(X)= Since we have E(X – µ)² = E(X² – 2µX + µ) = EX² – 2µEX + µ² = EX² – (EX)² the variance of X: Var(X)=.
Define a random process ✗(t) as follows: (1) X(t) assumes only one of two possible levels 1 or -1 at any time, (2) X(t) switches back and forth between its two levels randomly with time, (3) the number of level transitions in any time interval is a Poisson random variable, that is, the probability of exactly k transitions, when the average rate of transitions is λ, 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 levels at the start of any interval are equally probable. X(t) is usually called the random telegraph process. It is an example of a discrete random process. function o Seess (d) Discuss the stationarity of X(t).
Define a random process ✗(t) as follows: (1) X(t) assumes only one of two possible levels 1 or -1 at any time, (2) X(t) switches back and forth between its two levels randomly with time, (3) the number of level transitions in any time interval is a Poisson random variable, that is, the probability of exactly k transitions, when the average rate of transitions is λ, is given by [(t)/k!] exp(-λ), (4) transitions occurring in any time interval are sta- tistically independent of transitions in any other interval, and (5) the levels at the start of any interval are equally probable. X(t) is usually called the random telegraph process. It is an example of a discrete random process. function of Seess (c) What is E[X(t)]?
d) The random variable X has a Poisson distribution such that P(X= 0) = P(X = 1). Calculate P(X = 2).
1. (a) A continuous random variable X has a p. d. f. f (x) = 3x², 0 a) and (ii)P(X > b) = 0.5. (b) In the rolling of two dice, let X represents the total number on the upturned faces. Construct a table showing the probability associated with each and every possibility of the numbers. Calculate the distribution function of X. Also, calculate expected value of X.
2. Let X be a random variable with the Poisson distribution, parameter 2. Show that, for w = 1, 2, 3, ..., P(X > w) = P(Y < 2), where Y is a random variable having the gamma distribution with parameters w and 1.
If X₁, X₁, X3, ..., xn are random sample from a population with mean µ and variance o², then what is ε[(x₁ - μ)(X; -μ)] for ij, i = 1,2,3,..., n?

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