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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![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ec19564-4418-496c-9b50-60daa8944313%2Fd43673ed-ef9f-41bb-a442-2e53ebd118bf%2F6tlj4c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**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.
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