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Chapter 2: Problem 6

Let \(X_{1}, X_{2}\) have the joint pdf \(f\left(x_{1}, x_{2}\right)=1 / \pi, 0

Short Answer

Expert verified
The joint pdf of \(Y_{1}\) and \(Y_{2}\) is \(\frac{1}{\sqrt{Y_1 - Y_2^2}}\times \frac{1}{\pi}, 0<y_{1}<1, -\sqrt{y_{1}}<y_{2}<\sqrt{y_{1}}\).

Step by step solution

01

Define the transformation

To start with, let's define the given transformation: \(Y_{1}=X_{1}^{2}+X_{2}^{2}\) and \(Y_{2}=X_{2}\). In order to apply the Jacobian Transformation method, we need to find the inverse transformation from \(Y_1\) and \(Y_2\) back to \(X_1\) and \(X_2\). The inverse functions are: \(X_{1}= \sqrt{Y_{1} - Y_{2}^2}\) and \(X_{2}=Y_{2}\).
02

Calculate the Jacobian

Next, calculate the Jacobian determinant of the transformation. The Jacobian determinant is the determinant of the Jacobian matrix, which contains the partial derivatives of the inverse transformation. So we solve \(鈹侸鈹� = 鈹俓partial(X_{1}, X_{2})/\partial(Y_{1}, Y_{2})鈹俓). Here, \(鈹侸鈹� = |\sqrt{Y_1 - Y_2^2} / Y_1| = \frac{\sqrt{Y_1 - Y_2^2}}{Y_1}\).
03

Find the joint pdf of \(Y_{1}\) and \(Y_{2}\)

Use the transformation formula for joint density functions: \(f_{Y_{1},Y_{2}}(y_{1},y_{2}) = f_{X_{1},X_{2}}(x_{1}(y_{1},y_{2}),x_{2}(y_{1},y_{2})) | J |\). Substituting the values of \(x_{1}\), \(x_{2}\) and \(| J |\), by considering the range 0<\(x_{1}^{2}+x_{2}^{2}<1\), we see that the conditions \(0<Y_1<1\) and \(-\sqrt{Y_1}<Y_2<\sqrt{Y_1}\) would hold. So, \(f_{Y_{1},Y_{2}}(y_{1},y_{2}) = \frac{1}{\sqrt{Y_1 - Y_2^2}}\times \frac{1}{\pi}, 0<y_{1}<1, -\sqrt{y_{1}}<y_{2}<\sqrt{y_{1}}\).

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Most popular questions from this chapter

Let \(f_{1 \mid 2}\left(x_{1} \mid x_{2}\right)=c_{1} x_{1} / x_{2}^{2}, 0

Let \(\mathbf{X}=\left(X_{1}, \ldots, X_{n}\right)^{\prime}\) be an \(n\) -dimensional random vector, with the variancecovariance matrix given in display (2.6.13). Show that the \(i\) th diagonal entry of \(\operatorname{Cov}(\mathbf{X})\) is \(\sigma_{i}^{2}=\operatorname{Var}\left(X_{i}\right)\) and that the \((i, j)\) th off diagonal entry is \(\operatorname{Cov}\left(X_{i}, X_{j}\right)\).

If the correlation coefficient \(\rho\) of \(X\) and \(Y\) exists, show that \(-1 \leq \rho \leq 1\). Hint: Consider the discriminant of the nonnegative quadratic function $$ h(v)=E\left\\{\left[\left(X-\mu_{1}\right)+v\left(Y-\mu_{2}\right)\right]^{2}\right\\} $$ where \(v\) is real and is not a function of \(X\) nor of \(Y\).

Let \(X\) and \(Y\) have the joint \(\operatorname{pmf} p(x, y)=\frac{1}{7},(0,0),(1,0),(0,1),(1,1),(2,1)\), \((1,2),(2,2)\), zero elsewhere. Find the correlation coefficient \(\rho .\)

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