If the p.m.f. of $X$ is given by $f(x)=p(1-p)^{x-1}$ where $0<p<1$, $x \in J^{+}$, determine the cumulative distrubution function of $X$.

Question

The answer is $F(x)=1-(1-p)^{x}$ I don't understand how this might be achieved without Integration.

Is it possible for this question to be answered without Integration?

Thanks.

Answer 1

We know that $F(x) = P(X \leq x) = P(X=x) + P(X=x-1) + ...P(X=1) = \sum_{i=1}^x P(X=i)$.

Therefore

\begin{align*} F(x) &= \sum_{i=1}^x P(X=i) \\ &= \sum_{i=1}^x p(1-p)^{i-1} \\ &= p \cdot \sum_{i=1}^x (1-p)^{i-1} \\ &= p \cdot \frac{1-(1-p)^{x}}{1-(1-p)} = 1-(1-p)^x \end{align*}