Finding Joint PDF of Two Exponential Random Variables

In summary, to find the joint PDF of two random variables, we can simply multiply their individual PDFs together. In the given example, the joint PDF of X and Y is f_{X,Y}(x,y) = 12e^{-3x-4y}. Using this, we can then find the PDF of W=X+Y by integrating over the joint PDF. This can be done using the formula for the density of a sum of independent random variables, which can be found in the article on probability density functions.
  • #1
EugP
107
0
Can anyone tell me how to find the joint PDF of two random variables? I can't seem to find an explanation anywhere. I'm trying to solve a problem but I'm not sure where to go with it:

Y is an exponential random variable with parameter [tex]\lambda=4[/tex]. X is also an exponential random variable and independent of Y with [tex]\lambda=3[/tex].. Find the PDF [tex]f_W(w)[/tex], where [tex]W=X+Y[/tex].

I know that I simply use:

[tex]f_W(w) = \int\int (x+y) f_{X,Y}(x,y)dydx[/tex]

The problem is that I don't know how to find their joint PDF. I know their PDF's separately:


[tex]f_X(x)=\left\{\begin{array}{cc}3e^{-3x},&
x\geq 0\\0, & otherwise\end{array}\right.[/tex]

[tex]f_Y(y)=\left\{\begin{array}{cc}4e^{-4x},&
x\geq 0\\0, & otherwise\end{array}\right.[/tex]

Would this help me in anyway? Please help.
 
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  • #2
the joint density function is simply the product of the individual density functions
see here under independence:
http://en.wikipedia.org/wiki/Probability_density_function
in that article you also find the correct formula for the density of X+Y, what you have there seems to be the formula for E[X+Y] imho
 
  • #3
judoudo said:
the joint density function is simply the product of the individual density functions
see here under independence:
http://en.wikipedia.org/wiki/Probability_density_function
in that article you also find the correct formula for the density of X+Y, what you have there seems to be the formula for E[X+Y] imho

Yeah sorry I realized I made a mistake, and that link helped a lot. Thank you!
 

Related to Finding Joint PDF of Two Exponential Random Variables

What is a joint PDF?

A joint probability density function (PDF) is a mathematical function that describes the probability of a specific outcome occurring in multiple variables simultaneously. In other words, it shows the likelihood of two or more random variables taking on specific values at the same time.

What is the difference between a joint PDF and a marginal PDF?

A marginal PDF is the probability distribution of a single variable, while a joint PDF is the probability distribution of multiple variables. Marginal PDFs can be obtained from joint PDFs by integrating out the other variables.

How do you find the joint PDF of two exponential random variables?

To find the joint PDF of two exponential random variables, you can use the formula f(x,y) = λe^(-λx-λy), where λ is the parameter of the exponential distribution. This formula assumes that the two random variables are independent and have the same λ value.

What is the relationship between the joint PDF and the joint CDF?

The joint cumulative distribution function (CDF) is the integral of the joint PDF and represents the probability that the two random variables will take on values less than or equal to a given value. In other words, the joint CDF is the cumulative version of the joint PDF.

How can the joint PDF be used in practice?

The joint PDF can be used to calculate probabilities of specific combinations of values for multiple variables. This is useful in many fields, such as finance, engineering, and science, where multiple variables may affect the outcome of a certain event or process. It can also be used to determine the correlation between two variables, as well as to perform statistical analyses and make predictions.

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