3) Show that the exponential pdf, {x(x) = () ex12, x>0 has the "lack of memory"...
5. The Exponential(A) distribution has density f(x) = for x<0' where λ > 0 (a) Show/of(x) dr-1. (b) Find F(x). Of course there is a separate answer for x 2 0 and x <0 (c Let X have an exponential density with parameter λ > 0 Prove the 'Inemoryless" property: P(X > t + s|X > s) = P(X > t) for t > 0 and s > 0. For example, the probability that the conversation lasts at least t...
3. Let X be an exponential random variable with parameter 1 = $ > 0, (s is a constant) and let y be an exponential random variable with parameter 1 = X. (a) Give the conditional probability density function of Y given X = x. (b) Determine ElYX]. (c) Find the probability density function of Y.
a) Show that the exponential distribution has a lack of memory and the minimum property b) What are the characteristic properties of the Poisson input process? Discuss some of their limitations by giving examples from practical queueing systems. c) Suppose there are 2 types of customers arriving at the same server according to independent Poisson processes. Show that the aggregate arrival process is still a Poisson process.
5. Let X have exponential pdf λe_AE 0 when x > 0 otherwise with λ = 3. Let Y-LX). Find E(Y) and Var(Y)
The life X (in years) of a regulator of a car has the
pdf
32 f(3) = 83 -e-(2/8), 0<x< 0. (a) What is the probability that this regulator will last at least 5 years? (b) Given that it has lasted at least 5 years, what is the conditional probability that it will last at least another 5 years? (c) Suppose the replacement cost Y in dollars) after the regular dies is proportional to X and with mean $5,120. Find...
Q1) A-Random variable X has the following Probability Density Function (PDF) fr(x)= 부.lel s 3. (0, xl>3, A1-Show that fr (x) is a valid PDF. B- X is a uniform (-1,3) random variable. Let Y be the output of a clipping circuit with the input X such that Y - 80Q) where χ>0. , B1-Find P(Y-1). B2-Find P(Y 3). B3-Derive and plot the cumulative distribution function (CDF) of the random variable Y, Fy (). B4-What is the probability density function...
2. Let X be an exponential random variable with rate A > 0. In this problem you will show that X satisfies the memoryless property. Let s 2 0 and t > 0. Show that P(X > t + s| X > s) = e-M
1 x Suppose X has an exponential distribution, thus its pdf is given by fx (x) = 5e8,0 5x<0, 2> 0;0 0.w. a. Find E(X) b. Find E(X(X-1) c. Find Var (x)
3. Consider a continuous random variable X with pdf given by 0, otherwise This is called the exponential distribution with parameter X. (a) Sketch the pdf and show that this is a true pdf by verifying that it integrates to 1 (b) Find P(X < 1) for λ (c) Find P(X > 1.7) for λ : 1
I
am trying to understand why the conditional PDF f(X|X>1) is what
the answer the picture indicates given that P(X>1) = 1/e. Should
f(X|X>1) not equal e^(-x-1) instead given the exponential RV is
e^(-x)?
Example 5.20. Let X ~ Exponential(1). (a) Find the conditional PDF and CDF of X given X > 1. (b) Find E[X|X > 1). (c) Find Var(X|X > 1). Solution: (a) Let A be the event that X > 1. Then P(A) = S, ſed 1...