Averaging over the upper sum limit of a discrete function

[nikozm]

Hi,

Let the following function:

X = ∑^{L}_{k=1} f(k)/L, where f(k) is a continuous random function and L is a random discrete number. Both L and f(k) are non negative random variables. Thus, X is the average of f(k) with respect to L.

Is it right to say that X equals (or approximately) to ∑^{E[L]}_{k=1} f(k), where E[L] is the average discrete value of L ?

 

[mathman]

Not necessarily. Suppose f is constant for k < E(L) and increasing afterwards. The second sum is fixed, while the first sum would = second sum for L < E(L), but greater for L > E(L).