Finding Joint Density for Minimum of Independent Variables: $\min(A,B)$ Formula

In summary: C$ is the minimum of $A$ and $B$, it can only be greater than $c$ if both $A$ and $B$ are greater than $c$. This means that the probability of $A$ being greater than $c$ and the probability of $B$ being greater than $c$ must be multiplied together. Therefore, the expression for $f_C(c)$ is correct and can be simplified using the fact that $F_B(c)=1-e^{-\mu c}$ and $F_A(c)=1-e^{-\lambda c}$. In summary, the density for $C=\min(A,B)$ is $f_C(c)=2(\lambda+\mu)e^{-c(\lambda+\mu)}
  • #1
Jason4
28
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I have:

$f_A=\lambda e^{-\lambda a}$

$f_B=\mu e^{-\mu b}$

I need to find the density for $C=\min(A,B)$

($A$ and $B$ are independent).

Is this correct or utterly wrong?

$f_C(c)=f_A(c)+f_B(c)-f_A(c)F_B(c)-F_A(c)f_B(c)$

$=\lambda e^{-\lambda c}+\mu e^{-\mu c}-\lambda e^{-\lambda c}(1-e^{-\mu c})-(1-e^{-\lambda c})\mu e^{-\mu c}$

$=\lambda e^{-\lambda c}e^{-\mu c}+\mu e^{-\lambda c}e^{-\mu c}$

$=2(\lambda+\mu)e^{-c(\lambda+\mu)}$
 
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  • #2
Jason said:
I have:

$f_A=\lambda e^{-\lambda a}$

$f_B=\mu e^{-\mu b}$

I need to find the density for $C=\min(A,B)$

($A$ and $B$ are independent).

Is this correct or utterly wrong?

$f_C(c)=f_A(c)+f_B(c)-f_A(c)F_B(c)-F_A(c)f_B(c)$

You need to explain where this comes from.

Because we have two cases; \( A<B\) and \(A\ge B\) I would start:

$ \large f_C(c)=f_A(c)Pr(B>c|A=c)+Pr(A>c|B=c)) $

then independence reduces this to:

$ \large f_C(c)=f_A(c)Pr(B>c)+f_B(c)Pr(A>C) $

so:

\( \large f_C(c)=f_A(c)(1-F_B(c))+f_B(c)(1-F_A(c) \))

CB
 
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FAQ: Finding Joint Density for Minimum of Independent Variables: $\min(A,B)$ Formula

What is a joint density problem?

A joint density problem is a mathematical problem in statistics that involves finding the probability distribution of two or more random variables. It is used to determine the likelihood of two or more events occurring simultaneously.

How is a joint density problem different from a single variable probability problem?

A joint density problem involves multiple variables, while a single variable probability problem only involves one variable. In a joint density problem, the goal is to find the probability distribution of two or more variables, while in a single variable probability problem, the goal is to find the probability distribution of only one variable.

What is the formula for calculating joint density?

The formula for joint density is P(X,Y) = P(X) * P(Y|X), where P(X) and P(Y) are the marginal probabilities of X and Y, and P(Y|X) is the conditional probability of Y given X.

How is joint density used in real-life applications?

Joint density is used in many real-life applications, such as in finance, economics, and engineering, to model and analyze complex systems with multiple variables. It is also used in data analysis and machine learning to understand the relationships between different variables and make predictions.

What are some common methods for solving joint density problems?

There are several methods for solving joint density problems, including the joint distribution function, the joint probability density function, and the joint cumulative distribution function. Other methods include using conditional probability and the Bayes' rule to calculate the joint density. Additionally, computer software and statistical tools can be used to solve complex joint density problems.

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