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Chapter 2: Problem 5

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

Short Answer

Expert verified
The joint pdf of \(Y_{1}, Y_{2}\) is \( f_{Y_{1}, Y_{2}}(y_{1}, y_{2}) = f_{X_{1}, X_{2}}(y_{1} - y_{2}, y_{2})\) and the convolution formula is \(f_{Y_{1}}(y_{1}) = \int_{-\infty}^{\infty} f_{X_{1}, X_{2}}(y_{1}-y_{2}, y_{2}) dy_{2}\)

Step by step solution

01

Find Joint PDF of \(Y_{1}, Y_{2}\)

We have the transformation: \(Y_{1} = X_{1} + X_{2}\) and \(Y_{2} = X_{2}\). The inverse transformation will be: \(X_{1} = Y_{1} - Y_{2}\) and \(X_{2} = Y_{2}\). Delineate the Jacobian determinant of this transformation: \(J=\left|\frac{\partial(X_{1},X_{2})}{\partial(Y_{1},Y_{2})}\right| = \begin{vmatrix}\frac{\partial X_{1}}{\partial Y_{1}} &\frac{\partial X_{1}}{\partial Y_{2}} \\ \frac{\partial X_{2}}{\partial Y_{1}} &\frac{\partial X_{2}}{\partial Y_{2}}\end{vmatrix}=\begin{vmatrix}1&-1\\0&1\end{vmatrix}= 1\). The joint pdf of \(Y_{1}, Y_{2}\) is given by: \( f_{Y_{1}, Y_{2}}(y_{1}, y_{2}) = f_{X_{1}, X_{2}}(x_{1}, x_{2})|J|\) = \( f_{X_{1}, X_{2}}(y_{1} - y_{2}, y_{2})\)
02

Derive Convolution Formula

In order to compute \(f_{Y_{1}}(y_{1})\), we integrate \(f_{Y_{1}, Y_{2}}(y_{1}, y_{2})\) over all \(y_{2}\). Thus, \(f_{Y_{1}}(y_{1}) = \int_{-\infty}^{\infty} f_{Y_{1}, Y_{2}}(y_{1}, y_{2}) dy_{2} = \int_{-\infty}^{\infty} f_{X_{1}, X_{2}}(y_{1}-y_{2}, y_{2}) dy_{2}\). This is the convolution formula.

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Most popular questions from this chapter

2.6.2. Let \(f\left(x_{1}, x_{2}, x_{3}\right)=\exp \left[-\left(x_{1}+x_{2}+x_{3}\right)\right], 0

Find the probability of the union of the events \(a

Let \(X_{1}\) and \(X_{2}\) have the joint \(\operatorname{pdf} h\left(x_{1}, x_{2}\right)=8 x_{1} x_{2}, 0

Let the random variables \(X_{1}\) and \(X_{2}\) have the joint pmf described as follows: $$ \begin{array}{c|cccccc} \left(x_{1}, x_{2}\right) & (0,0) & (0,1) & (0,2) & (1,0) & (1,1) & (1,2) \\ \hline p\left(x_{1}, x_{2}\right) & \frac{2}{12} & \frac{3}{12} & \frac{2}{12} & \frac{2}{12} & \frac{2}{12} & \frac{1}{12} \end{array} $$ and \(p\left(x_{1}, x_{2}\right)\) is equal to zero elsewhere. (a) Write these probabilities in a rectangular array as in Example 2.1.3, recording each marginal pdf in the "margins." (b) What is \(P\left(X_{1}+X_{2}=1\right)\) ?

A person rolls a die, tosses a coin, and draws a card from an ordinary deck. He receives \(\$ 3\) for each point up on the die, \(\$ 10\) for a head and \(\$ 0\) for a tail, and \(\$ 1\) for each spot on the card \((\) jack \(=11\), queen \(=12\), king \(=13) .\) If we assume that the three random variables involved are independent and uniformly distributed, compute the mean and variance of the amount to be received.
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