Variation coefficient property

[Ad VanderVen]

TL;DR

Does a given property of a random variable also apply to the sum of that variable?

For a random variable T_i,

SD (T_i) / E (T_i) ≤ 1

with SD (T_i) = (Var (T_i))^1/2 and E (T_i) the expectation of T_i and Var (T_i) the variance of T_i. My question now is whether the following property then also applies. For any variable T,

SD (T) / E (T) ≤ 1

where T = T₁ + T₂ + ... + T_N and where N can be a fixed variable or random variable.

 

[mjc123]

Are the T_is independent?

 

[Ad VanderVen]

Not necessarily.

 

[mjc123]

I think the answer to your question is yes; S(T) ≤ E(T) whether or not the T_is are independent.

 

[Ad VanderVen]

That would be great, but how can you prove this?

 

[mjc123]

If S_i ≤ E_i for all i, then
∑S_i ≤ ∑E_i = E(T)
Now the question is, is S(T) ≤ ∑S_i?
V(T) = ∑V_i + 2∑∑'C_ij where C_ij is the covariance of T_i and T_j
V(T) = ∑V_i + 2∑∑'r_ijS_iS_j where r_ij is the correlation coefficient between T_i and T_j
This is a maximum when all the r's = +1. So
V(T) ≤ ∑S_i² + 2∑∑'S_iS_j
V(T) ≤ (∑S_i)²
S(T) ≤ ∑S_i
S(T) ≤ E(T)

 

[Stephen Tashi]

where T = T₁ + T₂ + ... + T_N and where N can be a fixed variable or random variable.

Suppose we have the case where $N$ is a random variable. Is $N$ known to be independent of the other random variables?

... and what would we mean by saying that $N$ is independent of a possibily infinite set of random variables ${T_1, T_2, ...}$ which are themselves not (necessarily) a set of independent random variables?