Question

Let $$ f(x, y)=(1 / 2 \pi) \exp \left[-\frac{1}{2}\left(x^{2}+y^{2}\right)\right]\left\\{1+x y \exp \left[-\frac{1}{2}\left(x^{2}+y^{2}-2\right)\right]\right\\} $$ where \(-\infty

Short answer

To compile a short answer, the integral of given function over all range (x,y) should be shown equal to 1 for it to be a joint pdf, and then marginal pdf's have to be obtained and shown that they are normal. The fact that the given joint pdf is not normal despite its marginal pdf's being normal serves as an exception to the expectation that joint pdf would be normally distributed if marginal pdf's are normal.

Step-by-step solution

Verify if \(f(x, y)\) is a joint pdf

To prove that \(f(x, y)\) is a joint pdf, the integral of it over all \(x\) and \(y\) should be equal to 1. The function is defined over all \(x\) and \(y\), hence we need to determine\n\n\[\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) dx dy.\]\n\nAfter integrating and calculating, if this integral equals to 1, we can say \(f(x, y)\) is a joint pdf.

Determine the marginal pdf's

The marginal pdf of \(X\) is given by \n\n\[f_{X}(x) = \int_{-\infty}^{\infty} f(x, y) dy.\]\n\nSimilarly, the marginal pdf of \(Y\) is given by \n\n\[f_{Y}(y) = \int_{-\infty}^{\infty} f(x, y) dx.\]\n\nEvaluate these integrals to determine the marginal pdf's.

Show marginal pdf's are normal

After calculating the marginal pdf's, show that they are normal pdf's by comparing them to the standard form of a normal pdf, which is \n\n\[f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{ - \frac{(x - \mu)^2}{2 \sigma^2}},\]\n\nwhere \(\mu\) is the mean (expectation) and \(\sigma\) is the standard deviation. If the marginal pdf's have the same form as a normal pdf, they are normal distributions.