Math Problem Statement
Q 2.5. Let X and Y be independent Exponential(λ) random variables. Let M = min{X, Y } and U = max{X, Y }.
- Compute the distribution function and probability density of M.
- Compute the distribution function and probability density of U.
- Compute the joint distribution function F(M,U) (m, u) of M and U, and thus deduce the joint density fM,U (m, u).
Solution
To tackle this problem, let’s delve into each part step by step.
1. Distribution Function and Probability Density of
Since and are independent exponential random variables with parameter :
- The cumulative distribution function (CDF) for an exponential random variable is given by:
The minimum has the CDF:
Thus, the probability density function (PDF) of is the derivative of the CDF:
2. Distribution Function and Probability Density of
For the maximum :
Thus, the PDF of :
3. Joint Distribution Function
To find the joint distribution :
This implies:
Thus, the joint PDF :
Calculating this leads to:
Summary
- ,
- ,
- leads to the joint density
Would you like to delve into any particular part or have more questions? Here are some suggestions:
- How do independent random variables influence the behavior of and ?
- Can you provide examples of applications for these distributions?
- What are the properties of exponential distributions?
- How do you compute moments for and ?
- How does the joint density relate to other probability distributions?
Tip: Familiarize yourself with the memoryless property of exponential distributions, as it simplifies many calculations.
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Exponential Distributions
Joint Distribution
Formulas
F_X(x) = 1 - e^{-λx}, x ≥ 0
F_M(m) = 1 - e^{-2λm}
f_M(m) = 2λe^{-2λm}
F_U(u) = (1 - e^{-λu})^2
f_U(u) = 2λe^{-λu}(1 - e^{-λu})
F(m, u) = F_U(u) - (1 - e^{-λm})^2
f_{M,U}(m, u) = 2λ^2 e^{-λ(m + u)}
Theorems
Memoryless Property of Exponential Distributions
Properties of Independent Random Variables
Suitable Grade Level
Grades 11-12
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