Moments and Moment-gen. function.

[SithV]

Hi people!

Im having problems with my Prob.Theor. assignment=(
I was hoping that u might be able to help me...

I have 2 problems that i ve no idea how to solve!Oo

Heres 1st one
We re given rand. var. X, its mean value U, the stand deviation S (sigma).
We need to show that E(z)=0 and var(z)=1
if the relation between X and Z is this eq. Z=X-U/S

2nd
Show if a rand. var. has the prob. density
f(x)=1/2*Exp[-lxl] -inf<X<inf
lxl-abs value
then its moment gen func. is
Mx(t)=1/1-t^2

im not sure about this one bu here's what i got
we re using the formula from the definition
and gettin this
1/2(Int[Exp[tx]*Exp[-lxl]) -inf<X<inf

but lxl=+-x

then we get 2 integrals
1/2(Int[Exp[tx]*Exp[-x])
and
1/2(Int[Exp[tx]*Exp[x])
both in -inf<X<inf

now if we integrate it we get
1/2Exp[x(t-1)]/t-1
and
1/2Exp[x(t+1)]/t+1
both in -inf<X<inf

whats next?=(

Would appreciate any help!=(

 

[mathman]

For your first problem z=(x-U)/S. Direct calculation will give you the results.

For your second problem, the ranges of integration of the split integrals are (-∞,0) for the +x integral and (0,∞) for the -x integral. Plug in x=0 and the resulting terms (be careful of signs) add to get the answer.

 

[SithV]

Ok, i think i got the second one.., but what do i do with 1/2 in front of 1/1+t and 1/1-t when i multiply them?

Now the to first one. You mean i need to integrate it? Like this: Int[Z*(X-U)/S], but what range do i pick?
And then var(Z) i can find from S^2=U'2-U^2
which is S=Sqrt[U'2-U^2]
where U^2 is E(Z)
and U'2=Int[(Z^2)*(X-U)/S]
but again what range?
Thank you!

 

[mathman]

1/(1-t) + 1/(1+t) = 2/(1-t²). So multiply by 1/2 to get your answer.

The range for the integration is the entire real line.
A simpler approach is as follows:
E[(X-U)/S] = {E(X) - U}/S = 0
E[({X-U)/S}²] = {E(X²) -2UE(X) +U²}/S² = 1

 

[blue_raver22]

Hi there! It sounds like you're working on some interesting problems in probability theory. Let's break down each problem and see if we can figure out a solution together.

For the first problem, we are given a random variable X with a mean value of U and a standard deviation of S. We need to show that the expected value of a transformed random variable Z, given by Z = (X-U)/S, is 0 and that its variance is 1.

To show that the expected value of Z is 0, we can use the definition of expected value, E(Z) = ∫Z*f(z)dz, where f(z) is the probability density function of Z. Since Z = (X-U)/S, we can substitute this into the definition to get E(Z) = ∫(X-U)/S * f(z)dz. Now, we know that the probability density function of Z is related to the probability density function of X through f(z) = f((z*S)+U). Substituting this in, we get E(Z) = ∫(X-U)/S * f((z*S)+U)dz. Since we have the mean and standard deviation of X, we can use these to calculate the probability density function of X and then substitute it into this equation. After some algebraic manipulation, we should be able to show that E(Z) = 0.

For the variance, we can use the formula Var(Z) = E[(Z-E(Z))^2]. Again, we can use the definition of expected value and the probability density function of Z to calculate this. After some algebraic manipulation, we should be able to show that Var(Z) = 1.

For the second problem, we are given a probability density function f(x) = 1/2 * Exp[-lxl] for -inf<X<inf and we need to find its moment generating function, Mx(t). To do this, we can use the definition of moment generating function, Mx(t) = E[Exp(tx)]. Again, we can use the definition of expected value and the probability density function to calculate this. After some algebraic manipulation, we should be able to show that Mx(t) = 1/(1-t^2).

I hope this helps! Remember, when working on math problems, it's always helpful to break them down step by step and use the definitions and formulas that you know. Best