Let X1 and X2 be continuous random variables with the joint...

Let $X_{1}$ and $X_{2}$ be continuous random variables with the joint probability density function $f_{X_{1}, X_{2}}\left(x_{1}, x_{2}\right),-\infty<x_{i}<\infty, i=1,2 .$ Let $Y_{1}=X_{1}+X_{2}$ and $Y_{2}=X_{2}$ (a) Find the joint pdf $f_{Y_{1}, Y_{2}}$. (b) Show that $$ f_{Y_{1}}\left(y_{1}\right)=\int_{-\infty}^{\infty} f_{X_{1}, X_{2}}\left(y_{1}-y_{2}, y_{2}\right) d y_{2} $$ which is sometimes called the convolution formula.

Let $X_{1}$ and $X_{2}$ be continuous random variables with the joint probability density function $f_{X_{1}, X_{2}}\left(x_{1}, x_{2}\right),-\infty<x_{i}<\infty, i=1,2 .$ Let $Y_{1}=X_{1}+X_{2}$ and
$Y_{2}=X_{2}$
(a) Find the joint pdf $f_{Y_{1}, Y_{2}}$.
(b) Show that
$$
f_{Y_{1}}\left(y_{1}\right)=\int_{-\infty}^{\infty} f_{X_{1}, X_{2}}\left(y_{1}-y_{2}, y_{2}\right) d y_{2}
$$
which is sometimes called the convolution formula.

Key Concepts

Marginalization and Convolution

Marginalization involves integrating out one or more variables from a joint probability density function to obtain the marginal density of a subset of the variables. The convolution formula, which arises from this process when dealing with sums of random variables, is integral in determining the distribution of a sum of independent or dependent random variables. It effectively sums the contributions from all possible values of one variable to yield the marginal density of the sum.

Jacobian and Change of Variables Technique

The Jacobian is a determinant used in the process of changing variables in a multivariate probability density function. It accounts for the distortion (scaling) of volume elements during the transformation, ensuring that the new density function is properly normalized. The change of variables technique by employing the Jacobian is essential in deriving the joint pdf of transformed random variables.

Transformation of Random Variables

Transformation of random variables involves substituting one set of random variables with another by applying a specified function or mapping. This is particularly useful when analyzing functions of random variables since it allows the derivation of the new variable's probability distribution by relating it to the original variable's distribution.

Joint Probability Density Function

A joint probability density function (pdf) describes the likelihood of two or more continuous random variables taking on specific values simultaneously. It is a function used to model the probability distribution for a system of random variables, and its integral over any subset of the support gives the probability that the random variables fall within that region.

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