Question

Relation between cumulative and moments in probability theory

Answer 1

Here's a sketch for the continuous case:

For any nonnegative continuous real random variable XX and any integer r≥1r≥1,

Xr=∫X0rxr−1dx=∫∞0rxr−1[X>x]dxXr=∫0Xrxr−1dx=∫0∞rxr−1[X>x]dx

where

[X>x]={10if X>xif X≤x.[X>x]={1if X>x0if X≤x.

Therefore, using Tonelli's theorem and the fact that E[X>x]=P(X>x)E[X>x]=P(X>x),

E(Xr)=r∫∞0xr−1P(X>x)dx.E(Xr)=r∫0∞xr−1P(X>x)dx.

Now, for any continuous random variable XX (not necessarily nonnegative), we have X=Y−ZX=Y−Z, where Y=X+Y=X+ and Z=X−Z=X− are the positive and negative parts of XX. Then, since YZ=0YZ=0, the Binomial Theoremgives

Xr=(Y−Z)r=Yr+(−Z)rXr=(Y−Z)r=Yr+(−Z)r

and because both YY and ZZ are nonnegative random variables,

E(Xr)=E(Yr)+(−1)rE(Zr)=r∫∞0yr−1P(Y>y)dy+(−1)rr∫∞0zr−1P(Z>z)dz=r∫∞0yr−1P(X>y)dy+(−1)rr∫∞0zr−1P(X<−z)dz=r∫∞0xr−1(P(X>x)+(−1)rP(X<−x))dx=r∫∞0xr−1(1−F(x)+(−1)rF(−x))dx.

I hope it will help you dear...

Answer 2

Here's a sketch for the continuous case:

For any nonnegative continuous real random variable XX and any integer r≥1r≥1,

Xr=∫X0rxr−1dx=∫∞0rxr−1[X>x]dxXr=∫0Xrxr−1dx=∫0∞rxr−1[X>x]dx

where

[X>x]={10if X>xif X≤x.[X>x]={1if X>x0if X≤x.

Therefore, using Tonelli's theorem and the fact that E[X>x]=P(X>x)E[X>x]=P(X>x),

E(Xr)=r∫∞0xr−1P(X>x)dx.E(Xr)=r∫0∞xr−1P(X>x)dx.

Now, for any continuous random variable XX (not necessarily nonnegative), we have X=Y−ZX=Y−Z, where Y=X+Y=X+ and Z=X−Z=X− are the positive and negative parts of XX. Then, since YZ=0YZ=0, the Binomial Theoremgives

Xr=(Y−Z)r=Yr+(−Z)rXr=(Y−Z)r=Yr+(−Z)r