I was solving the following question:
Find the CDF of a Binomial Distribution.
For some reason, I missed the fact that the CDF of a Binomial Distribution is simply
$$F(x) = \sum_{i=0}^x \binom{n}{i}p^x(1-p)^{n-i}$$ and simply did the following
\begin{align*} P(X \leq x) &= p(0) + p(1) + p(2) + \ldots + p(x) \\ &= \binom{n}{0} p^0 (1-p)^n + \binom{n}{1} p^1 (1-p)^{n-1} + \binom{n}{2} p^2 (1-p)^{n-2} + \ldots + \binom{n}{x} p^x (1-p)^{n-x} \\ &= \sum_{t=0}^n \binom{n}{t} (-p)^{n-t} \sum_{i=0}^x \binom{n}{x} \left(\frac{p}{1-p}\right)^i \quad \text{for } 0 < p < 1 \end{align*}
I applied the Binomial Theorem to express $(1-p)^n$ as a sum, viz. $$(1-p)^n = \sum_{t=0}^n \binom{n}{t} (-p)^{n-t}$$ and since it is constant inside the first sum, I took it out as a constant. Is this approach mathematically valid/correct?