Variation coefficient property

  • #1

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.
 

Answers and Replies

  • #2
mjc123
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Are the Tis independent?
 
  • #4
mjc123
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I think the answer to your question is yes; S(T) ≤ E(T) whether or not the Tis are independent.
 
  • #5
That would be great, but how can you prove this?
 
  • #6
mjc123
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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)
 
  • #7
Stephen Tashi
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where T = T1 + T2 + ... + TN 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?
 

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