二元正态： $\mathbb{E}[e^X\mid Y=y_0]$

二元正态分布计算

专题: Algorithmic Programming / 算法编程
难度: L4
来源: QuantQuestion

题目详情

Assume that $X$ and $Y$ are bivariate normal random variables with mean 0 and covariance matrix $\Sigma = \left[ \begin{array}{cc}1 & \rho \\ \rho & 1 \end{array} \right]$ . Evaluate $\mathbb{E}\left[e^{X} \mid Y = y_{0}\right]$

解析

对 $(X,Y)$ 二元正态， $X\mid(Y=y_0)$ 仍为正态：

X\mid Y=y_0\sim N(\rho y_0,\ 1-\rho^2).

因此

\mathbb{E}[e^X\mid Y=y_0]=\exp\left(\rho y_0+\frac{1-\rho^2}{2}\right).