# Normal distribution and extremal value

1. Jan 28, 2006

### faruk

Normal distribution.

What is the value of sigma (dispersion) for maximal probability P(1<x<2) ?

Excel calculation: sigma is about 1.471. But what would be an analytical solution?

2. Jan 29, 2006

### Tide

HINT: Calculate the derivative with respect to $\sigma$ of the integral of the distribution function over the given interval.

3. Jan 29, 2006

### faruk

That's exact the problem.
exp(-x²) belongs to the unintegratable functions. It's the cause we use the table of the normal distribution probability.

4. Jan 29, 2006

### HallsofIvy

Staff Emeritus
Your precise wording is wrong. exp(-x2) is integrable- it's integral just doesn't happen to be an elementary function. (Actually, its integral is the error function because that's how the error function is defined!)

But you don't need to know the function itself you only need to know its derivative. What is this derivative:
$$\frac{d}{dx}\left(\int_a^x e^{-t^2}dt\right)$$

Hint: What is this derivative:
$$\frac{d}{dx}\left(\int_a^x f(t)dt\right)$$