- #1

QuantumBunnii

- 15

- 0

## Homework Statement

Consider the gaussian distribution

ρ(x) = Aexp[(-λ^2)(x-a)^2] ,

where A, a, and λ are positive real constants.

(a) Find A such that the gaussian distribution function is normalized to 1.

(b) Find <x> (average; expected value) , <x^2>, and σ (standard deviation).

(c) Sketch the graph of ρ(x)

## Homework Equations

Gaussian integrals (integrated from 0 to infinity):

∫x^(2n)exp[(-x^2)/a^2]dx = [itex]\sqrt{\pi}[/itex][itex]\frac{(2n)!}{n!}[/itex]([itex]\frac{a}{2}[/itex])^(2n+1)

∫x^(2n+1)exp[(-x^2)/a^2]dx = [itex]\frac{n!}{2}[/itex]a^(2n+2)

## The Attempt at a Solution

(a) This part didn't give me any problems (I think), but I would like to make sure the general methodology is correct.

In order to normalize the distribution function to 1, you would merely set the integration over all space equal to 1:

∫Aexp[(-λ^2)(x-a)^2]dx = 1 (all space)

Since the only variable that differs from the given gaussian integral is the (x-a) term in the exponent, we can change the variable to, say, y = x-a and get dy = dx. Plugging everything in straight from the gaussian integral and multiplying everything by 2 (the given integrals are from 0 to inf., whereas this is from negative inf. to positive inf.), A is readily obtained as [itex]\frac{2λ}{\sqrt{\pi}}[/itex] .

(b) This is where I start to have trouble.

The average, <x> is given as follows:

∫xρ(x) dx = ∫xAexp[(-λ^2)(x-a)^2]dx (all space)

and this is as far as I can get. My problem is essentially one of mathematics: I don't know how to solve this integral. Just as in the previous problem, it is extremely close to the given gaussian integral, expect that-- in this case-- we can't employ a change of variables because we have the 'x' sitting in front of everything. The same will, of course, apply to <x^2>, and-- since σ = <x^2> - <x>^2-- this inhibits me from moving any further.

(c) is again a problem which arises from my lack of mathematical competence. I'm simply not sure how to sketch gaussian distribution functions. Could it be that the average (or, 'expected') value represents the peak of this function? (If so, I don't understand why-- as the average value is not necessarily the most probable value). And perhaps the standard deviation σ represents something akin to the full-width-at-half-maximum?

Or maybe I simply need to plug in a variety of points to get a general idea?

Thanks for any help~

P.S. Sorry about the formatting of the equations. I'm not sure how to make them look any better. :p