0

I have infill criterion, or sometimes called acquisition function and I want to calculate it. $\mathbb{EI}[x] = \mathbb{E}[I(x)]$, where $I(x) := max\{y_{min} - Y(x), 0\}$ and also $Y(x) \sim N(\mu, \sigma^2)$

Solution (not finished)

1) using this post Expected value of maximum of two random variables from uniform distribution

I calculated following : \begin{equation} \mathbb{EI}[x] = \int_0^{\infty} [1 - \mathbb{P}(y_{min} - Y(x) \leq z)]dz \end{equation}

But don't know how to proceed further, I would appreciate any suggestion/tips/advises...

Thank you in advance

Koval Boris
  • 74

1 Answers1

1

We have $I = f(Y)$ for $$f(y) = \max\{y_{\min} - y,0\} = (y_{\min} - y) 1_{y_{\min} \ge y}$$ where $1$ denotes the indicator function. Hence $$E[I] = E[f(Y)] = \int_{-\infty}^\infty f(y) g(y) dy$$ where $g(y)$ is the density of a normal distribution

Simplifying we get: $$\begin{align*} E[I] &= \int_{-\infty}^\infty f(y) g(y) dy \\ &= \int_{-\infty}^{y_\min} (y_\min - y) g(y) dy \\ &= y_\min \int_{-\infty}^{y_\min} g(y) dy - \int_{-\infty}^{y_\min} yg(y) dy \\ &= y_\min P(Y \le y_\min) - \int_{-\infty}^{y_\min} yg(y) dy \end{align*}$$

And the second integral you can easily solve using substitution.

Gono
  • 5,598