*Originally posted by Acolyte*

**(Apologies for the nature of this puzzle. It's generally seen as a good thing to show 17-year-old maths students who want something a little harder than standard A-level questions.)
**

The [b]standard normal distribution is defined by the probability density function

f(x) = K*exp(-x^2), where K is constant. Find K.[/b]

(Disclaimer: I don't know any stats beyond the AP curriculum, which consists of a lot of messing about with calculators, so I didn't even know the form of the normal distribution until Acolyte just said it. For shame!)

Well, since it's a continuous probability distribution, integrating it from zero to infinity wrt x should give 1, so I guess:

K = [INTEGRAL(0 to infinity) exp(-x^2) dx]^-1

Let's call that integral I, 'cause it's typographically mean. We have two options here, namely to integrate the Maclaurin series of the integrand term-by-term, which would give an answer in terms of gamma(1/2), which won't do a lot of good. Past experience is making a double integral in polar coordinates look like the sexy thing to do, so, noting that the integrand is always positive, we will find I^2 and take the positive square root:

I^2 = INT(0 to inf)INT(0 to inf) exp(-(x^2 + y^2)) dydx

= INT(0 to pi/2)INT(0 to inf)r*exp(-r^2) drdq

= pi/2*INT(0 to inf)r*exp(-r^2)dq

= pi/4

So that I = pi^(1/2)/2 and thus K = 2*pi^(-1/2). That right?