# Probability : joint density function of 3 Normal Distributions

• kkirtac
In summary, X1, X2, and X3 are independent gaussian random variables and Y1, Y2, and Y3 are given. The joint pdf of Y1, Y2, and Y3 can be found by convolving the individual pdfs of X1, X2, and X3. For Y2 and Y3, if the random variables are gaussian and centered, their pdfs will be the convolution of the respective pdfs of X1, X2, and X3. For more information, the book of Papoulis can be referenced.
kkirtac
X1, X2, X3 are independent gaussian random variables.
Y1 = X1+X2+X3
Y2 = X1-X2
Y3 = X2-X3
are given. What is the joint pdf of Y1,Y2 and Y3 ?

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kkirtac said:
X1, X2, X3 are independent gaussian random variables.
Y1 = X1+X2+X3
Y2 = X1-X2
Y3 = X2-X3
are given. What is the joint pdf of Y1,Y2 and Y3 ?

There seems to be some info missing here. Are X1, X2, and X3 independent? Are they discrete or continuous? Either way, what values can X1, X2, and X3 take? All this information is necessary in order to find the joint pdf of Y1, Y2, and Y3.

Dear,
If you are always watching the post.
the pdf of Y1 is the convolution of the three pdfs of the three random variables (X1, X2 and X3).
for Y2 and Y3, if the random variables are gaussian and centered (mean = 0) then pdf(X) = pdf(-X) and thus for pdf(Y2) = convolution of pdf(X1) and pdf(X2) while pdf(Y3) = convolution of pdf(X2) and pdf(X3).
Actually, I address you to the great book of Papoulis where you can find (for sure) the answer to your wondering .
Cheers
Manar

## 1. What is a joint density function?

A joint density function is a mathematical function that describes the probability of multiple random variables taking on specific values simultaneously. In other words, it shows the likelihood of two or more events occurring together.

## 2. How is a joint density function different from a single variable density function?

A joint density function takes into account the probabilities of multiple variables, while a single variable density function only considers the probability of one variable. In other words, a joint density function is a multivariate function, while a single variable density function is univariate.

## 3. What are the properties of a joint density function?

A joint density function must satisfy three properties: it must be non-negative for all values of the variables, the integral over all possible values must equal 1, and it must be continuous everywhere except for a finite number of points.

## 4. How do you calculate the probability from a joint density function?

The probability of an event occurring can be calculated by taking the integral of the joint density function over the region corresponding to that event. This integration can be done using calculus.

## 5. Can a joint density function be used for more than three normal distributions?

Yes, a joint density function can be used for any number of normal distributions. However, as the number of distributions increases, the calculations become more complex and may require advanced mathematical techniques.

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