Question

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.

Added by Michael W.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Adi S

05/29/2023

Step 1

The transformation is given by: $$ Y_1 = X_1 + X_2 \\ Y_2 = X_2 $$ Now, we need to find the inverse transformation from $(Y_1, Y_2)$ to $(X_1, X_2)$: $$ X_1 = Y_1 - Y_2 \\ X_2 = Y_2 $$ Show more…

Show all steps

Thanks for your feedback!

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