# Variation coefficient property

## Summary:

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

## Main Question or Discussion Point

For a random variable Ti,

SD (Ti) / E (Ti) ≤ 1

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

SD (T) / E (T) ≤ 1

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

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mjc123
Homework Helper
Are the Tis independent?

Not necessarily.

mjc123
Homework Helper
I think the answer to your question is yes; S(T) ≤ E(T) whether or not the Tis are independent.

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

mjc123
Homework Helper
If Si ≤ Ei for all i, then
∑Si ≤ ∑Ei = E(T)
Now the question is, is S(T) ≤ ∑Si?
V(T) = ∑Vi + 2∑∑'Cij where Cij is the covariance of Ti and Tj
V(T) = ∑Vi + 2∑∑'rijSiSj where rij is the correlation coefficient between Ti and Tj
This is a maximum when all the r's = +1. So
V(T) ≤ ∑Si2 + 2∑∑'SiSj
V(T) ≤ (∑Si)2
S(T) ≤ ∑Si
S(T) ≤ E(T)

Stephen Tashi