Let x have an exponential distribution with λ = 1. Find the probability. (Round your answer to four decimal places. P(x > 2)
Let x have an exponential distribution with λ = 1. Find the probability. (Round your answer to four decimal places. P(x > 2)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![**Problem Statement:**
Let \( x \) have an exponential distribution with \( \lambda = 1 \). Find the probability. (Round your answer to four decimal places.)
\[ P(1 < x < 6) \]
**Answer Box:**
[Input box for response]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9175f245-2e1a-4def-805b-da74e3897838%2F042ffdca-6d4a-4497-b906-c44617289dd1%2Fyzurtth_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( x \) have an exponential distribution with \( \lambda = 1 \). Find the probability. (Round your answer to four decimal places.)
\[ P(1 < x < 6) \]
**Answer Box:**
[Input box for response]
![**Problem:**
Let \( x \) have an exponential distribution with \( \lambda = 1 \). Find the probability. (Round your answer to four decimal places.)
\[ P(x > 2) \]
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**Solution:**
To find the probability \( P(x > 2) \), use the formula for the cumulative distribution function (CDF) of an exponential distribution:
\[ P(x > a) = 1 - F(a) = 1 - (1 - e^{-\lambda a}) = e^{-\lambda a} \]
For this problem, \( \lambda = 1 \) and \( a = 2 \):
\[ P(x > 2) = e^{-1 \times 2} = e^{-2} \]
Calculating \( e^{-2} \), we get:
\[ P(x > 2) \approx 0.1353 \]
Therefore, the probability that \( x \) is greater than 2 is approximately **0.1353**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9175f245-2e1a-4def-805b-da74e3897838%2F042ffdca-6d4a-4497-b906-c44617289dd1%2Fts4253j_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Let \( x \) have an exponential distribution with \( \lambda = 1 \). Find the probability. (Round your answer to four decimal places.)
\[ P(x > 2) \]
---
**Solution:**
To find the probability \( P(x > 2) \), use the formula for the cumulative distribution function (CDF) of an exponential distribution:
\[ P(x > a) = 1 - F(a) = 1 - (1 - e^{-\lambda a}) = e^{-\lambda a} \]
For this problem, \( \lambda = 1 \) and \( a = 2 \):
\[ P(x > 2) = e^{-1 \times 2} = e^{-2} \]
Calculating \( e^{-2} \), we get:
\[ P(x > 2) \approx 0.1353 \]
Therefore, the probability that \( x \) is greater than 2 is approximately **0.1353**.
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