# PDF for minimum and maximum of n independent Gaussian random variables

Stimulated by the problem Let $Z\sim N(0,1)$ be a random variable, then $E[\max\{Z,0\}]$ is? I came up with this problem:

Let $x_i, i=1..n$ be $n$ independent random variables $\sim N(0,1)$.

1) Caculate the PDF of the minimum and maximum of the $x_i$, respectively, for $n = 2$ and $n = 3$
2) Calculate the first and second moments
3) generalize the results to any $n$

Here's my start. Let $x$ and $y$ be the random variables. The PDF of $x$ is

$$f(x) = \frac{e^{-\frac{x^2}{2}}}{\sqrt{2 \pi }}$$

and similarly for $y$.

For the easy of writing we combine $min$ and $max$ into $ext$

Let

$$t=ext (x,y)$$

The PDF of $t$ is given by

$$fExt(t)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x) f(y) \delta (t-ext (x,y))dydx$$

I solved the problem for n = 2, and partly for n = 3. In order not to spoil your fun I shall provide my solution depending on your answers.

• The min and max cases should be symmetrical. For the minimum case, see: stats.stackexchange.com/questions/77692/… ............... For the maximum case, see math.stackexchange.com/questions/473229/… – wolfies Jan 20 '17 at 16:39
• @wolfies Thank you. But your reference on the case of the maximum is not answering my question as it treats the expected value of the maximum, and I was asking for the pdf. – Dr. Wolfgang Hintze Jan 20 '17 at 23:43
• @wolfies Thank you for the interesting references. And, please forget my first comment. – Dr. Wolfgang Hintze Feb 2 '17 at 15:33

I have found for the PDFs the following expressions.

for the minimum

$$f_{min}(n,t)=\frac{n}{\sqrt{2 \pi }} \exp \left(-\frac{t^2}{2}\right) \left(\frac{1}{2} \left(1-\text{erf}\left(\frac{t}{\sqrt{2}}\right)\right)\right)^{n-1}$$

for the maximum

$$f_{max}(n,t)=\frac{n}{\sqrt{2 \pi }} \exp \left(-\frac{t^2}{2}\right) \left(\frac{1}{2} \left(1+\text{erf}\left(\frac{t}{\sqrt{2}}\right)\right)\right)^{n-1}$$

Reference to earlier work

For the pdf of the minimum see the answer of wolfies to https://stats.stackexchange.com/questions/77692/expected-value-of-minimum-order-statistic-from-a-normal-sample

For the maximum case there are related references which, however, give the expected value rather than the PDF

Expected value for maximum of n normal random variable

Expectation of the maximum of gaussian random variables