Hi, I can't see how the indexing is working in this derivation. The question is: Show that the cdf for a geometric random variable is given by F_{X}(t) = P(X \leq t) = 1 - (1-p)^{[t]}, where [t] denotes the greatest integer in t, t\geq 0.

The derivation is given as F_{X}(t) = P(X \leq t) = p \sum_{s=o}^{[t]}(1-p)^{s}

But \sum_{s=o}^{[t]}(1-p)^{s} = \frac{1-(1-p)^{[t]}}{1-(1-p)} =\frac{1-(1-p)^{[t]}}{p}

From which the result follows.

However, the sum of a the first n terms in a geometric series is: \sum_{k=0}^{n-1}ar^k = a \cdot \frac{1-r^n}{1-r}

So if we are saying that s=k, a = 1, r = (1-p) and n-1 = [t]

Then I can't see how this makes sense since it seems we are summing over [t]+1 terms, from s = 0 to s = [t]

Or in other words, if  n-1 = [t], why isn't the closed form of the sum over s as follows: \frac{1-(1-p)^{[t]+1}}{1-(1-p)} ?

Thanks in advance for any insights. MD