Question

Let \(Y_{1}

Short answer

The expectation \(E(Y_{1})\) is \(-\sigma / \sqrt{\pi}\) and the covariance of \(Y_{1}\) and \(Y_{2}\) can be calculated using the formula for covariance and the expectation values of \(Y_{1}\) and \(Y_{2}\).

Step-by-step solution

Calculate \(E(Y_{1})\) using joint pdf

The joint pdf of \(Y_{1}\) and \(Y_{2}\) can be given as follows considering \(N(0, \sigma^2)\): \[f_{Y_{1}, Y_{2}}(y_{1}, y_{2}) = \frac{{2e^{-\frac{{y_{1}^{2} + y_{2}^{2}}}{{2\sigma^{2}}}}}}{\sqrt{2\pi\sigma^{2}}}. \] Using this joint pdf, the expectation \(E(Y_{1})\) can be calculated as: \[E(Y_{1}) = \int_{-\infty}^{y_{2}}y_{1}f_{Y_{1}, Y_{2}}(y_{1}, y_{2})dy_{1}, \] which simplifies to \[E(Y_{1}) = \int_{-\infty}^{y_{2}}\frac{{-2y_{1}e^{-\frac{{y_{1}^{2} + y_{2}^{2}}}{{2\sigma^{2}}}}}}{\sqrt{2\pi\sigma^{2}}}dy_{1}. \] Integrating and simplifying gives \(E(Y_{1}) = -\sigma / \sqrt{\pi}\).

Calculate covariance of \(Y_{1}\) and \(Y_{2}\)

Covariance of \(Y_{1}\) and \(Y_{2}\) can be given as \[Cov(Y_{1}, Y_{2}) = E(Y_{1}Y_{2}) - E(Y_{1})E(Y_{2}).\] Since \(Y_{1}\) and \(Y_{2}\) are from \(N(0, \sigma^2)\), it is known that \(E(Y_{1}) = E(Y_{2})\), which makes their covariance to be \[Cov(Y_{1}, Y_{2}) = E(Y_{1}Y_{2}) - [E(Y_{1})]^2.\] Simplifying this gives the covariance of \(Y_{1}\) and \(Y_{2}\).