# Normal and Laplace moments

#### Tinkerz

I am trying to break this down very simply, some where I am not reading this correcting

u is the mean, sum of the numbers/count of the numbers
t= is the count of the number??
e= the exponential

if I then write the formula in excel, Exp(t*mean) the number is very large.

what is t in this formula?

I am missing something

Normal N(μ, σ2) = Laplace L(μ, b) = Moment-generating function - Wikipedia, the free encyclopedia

Thank you

#### drumist

$$\displaystyle t$$ does not really have a physical meaning. You can sort of think of it as a dummy variable.

The reason it is there is because of the properties that arise when you set it up that way.

From the series expansion, we can see that:

$$\displaystyle e^{tx} = 1 + tx + \frac{t^2x^2}{2!} + \frac{t^3x^3}{3!} + \cdots \implies \left[e^{tx}\right]_{t=0} = 1$$

$$\displaystyle \frac{d}{dt} \left( e^{tx} \right) = x + x^2t + \frac{x^3t^2}{2!} + \cdots \implies \left[ \frac{d}{dt} \left( e^{tx} \right) \right]_{t=0} = x$$

$$\displaystyle \frac{d^2}{dt^2} \left( e^{tx} \right) = x^2 + x^3t + \frac{x^4t^2}{2!} + \cdots \implies \left[ \frac{d^2}{dt^2} \left( e^{tx} \right) \right]_{t=0} = x^2$$

(etc.)

So the $$\displaystyle n \mbox{th}$$ derivative of $$\displaystyle e^{tx}$$ evaluated at $$\displaystyle t=0$$ is $$\displaystyle x^n$$

This turns out to be convenient because the $$\displaystyle n \mbox{th}$$ moment of a density function $$\displaystyle f(x)$$ is

$$\displaystyle \int_{-\infty}^{\infty} x^n f(x) \, dx$$

So if we take

$$\displaystyle M_X(t) = E \left[e^{tX}\right]=\int_{-\infty}^{\infty} e^{tx} f(x) \, dx$$

we can basically isolate one of the moments by taking the proper number of derivatives and setting $$\displaystyle t=0$$.

#### Tinkerz

So the derivative of evaluated at is we can basically isolate one of the moments by taking the proper number of derivatives and setting .

I would like to isolate two of the moments for a laplace moment

Can we do the first 2? b will be 3,

can we do it long hand so i understand the formula

thanks