# Properties of cumulative distribution function

$F(x)=\begin{cases} 0 & x < 0 \\ x^2 & 0 \le x < \frac{1}{2} \\ \alpha & x = \frac{1}{2} \\ 1 - 2^{-2x} & x > \frac{1}{2} \end{cases}$

1. Find $\text{Pr}\left (\frac{1}{4} < X \le \frac{3}{4}\right)$.
2. Find $\alpha$.

1. $P\left(\tfrac14<X\leq \tfrac34\right)=P\left(X\leq \tfrac34\right)-P\left(X\leq \tfrac14\right)$.
2. A cumulative distribution function is right-continuous. That is $F(x)=\lim_{y\downarrow x}F(y)$. This enables you to determine $\alpha=F(\tfrac12)$.