Asked by Kelsey Fratello on Sep 28, 2024
Verified
If the random variable X is exponentially distributed with parameter λ = 2,then the probability that X is between 1 and 2 equals the probability that X is between 2 and 3.
Exponentially Distributed
A probability distribution characterized by its constant rate of decay or growth, often used in the study of time until some specific event.
- Apply the memoryless property of exponential distributions to solve problems.
Verified Answer
YC
Yanna Caldwell1 day ago
Final Answer :
False
Explanation :
The probability that a continuous random variable falls within a particular interval is equal to the integral of the probability density function over that interval. The probability density function for an exponential distribution with parameter λ is f(x) = λe^(-λx), therefore the probability that X is between 1 and 2 is:
∫(from 1 to 2)λe^(-λx)dx = e^(-2) - e^(-4) ≈ 0.218
The probability that X is between 2 and 3 is:
∫(from 2 to 3)λe^(-λx)dx = e^(-4) - e^(-6) ≈ 0.0498
These probabilities are not equal, so the statement is false.
∫(from 1 to 2)λe^(-λx)dx = e^(-2) - e^(-4) ≈ 0.218
The probability that X is between 2 and 3 is:
∫(from 2 to 3)λe^(-λx)dx = e^(-4) - e^(-6) ≈ 0.0498
These probabilities are not equal, so the statement is false.
Learning Objectives
- Apply the memoryless property of exponential distributions to solve problems.
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