**Exponential Distribution and the Memoryless Property****Question 3: Exponential Distribution has a memoryless property.** Intuitively, this means that the probability of customer service answering your call (assuming the waiting time is exponential) in the next 10 minutes is the same, regardless of whether you have waited an hour on the line or have just picked up the phone. Formally, if \( X \sim \text{exponential}(\lambda) \), the probability density function is given by \( f(x) = \lambda \exp(-\lambda x) \). Given two positive numbers, \( t \) and \( s \), use the definition of conditional probability to show that:\[P(X > t + s \mid X > t) = P(X > s).\]**Hint:** First, find the cumulative distribution function (CDF) of \( X \) and note that:\[P(X > t + s \cap X > t) = P(X > t + s).\]This property illustrates the lack of memory in the exponential distribution, meaning past waiting times do not influence the probability of future waiting times, which is characteristic of the exponential distribution.

X~Exponential(λ)Then,f(x)=λe-λx,x>0We know,P(X≤x)=∫0xf(z)dz=∫0xλe-λzdzLet…

Q3. Exponential Distribution has a memoryless property. Intuitively, it means that the probability of customer service answering you call (assuming waiting time is exponential) in the next 10 mins is the same, no matter if you have waited an hour on the line or just picked up the phone. Formally, if X ~ exponential(A), f(x) = A exp(- Ax), and t and s are two positive numbers, use the definition of conditional probability to show that P(X > t +s| X > t) = P(X > s). Hint: Find the cdf of X first, and note that P(X >t+snX>t) = P(X >t + s)

Q3. Exponential Distribution has a memoryless property. Intuitively, it means that the probability of customer service answering you call (assuming waiting time is exponential) in the next 10 mins is the same, no matter if you have waited an hour on the line or just picked up the phone. Formally, if X ~ exponential(A), f(x) = A exp(- Ax), and t and s are two positive numbers, use the definition of conditional probability to show that P(X > t +s| X > t) = P(X > s). Hint: Find the cdf of X first, and note that P(X >t+snX>t) = P(X >t + s)

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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