Gaussian Expectation, using maximum

Question

Let $(A,B)$ have a standard Gaussian distribution on $\mathbb{R}^2$. Find, $\mathbb{E}[max(2.5A+B, A + 2.5Y)]$

I thought that this could first be rewritten as $\mathbb{E}(2.5A+B|A>B)*\mathbb{P}(A>B) + \mathbb{E}(A+2.5B|A<B)*\mathbb{P}(A<B)$.

However, this has not made the calculation any easier.

Answer 1

Here is an actual simplification:

• If $(A,B)$ are independent Gaussians (with variance $1$), then $(X=A+B,Y=A-B)$ are also independent Gaussians (with variance $2$).
• So you can rewrite \begin{align} \max(2.5A+B, A+2.5B)&=\max(1.75X+0.75Y,1.75X-0.75Y) \\ &=0.75X + \max(0.75Y,-0.75Y) \\ &=0.75X + 0.75|Y| \\ \end{align}
• Now the expectation value of $X$ is zero. And $|Y|$ has a half-normal distribution, which has expecation value $\frac{\sigma\sqrt{2}}{\sqrt{\pi}}$ in general, and here we have $\sigma^2=2$. Therefore the complete answer is \begin{align} 0.75\frac{\sqrt{2}\sqrt{2}}{\sqrt{\pi}}=\frac{3}{2\sqrt{\pi}} \end{align}

Hope this helps.