109. The random variable $X$ is given by the probability density function:

$$
f(x)=\left\{\begin{array}{ccc}
0 & \text { if } & x \leq \frac{\pi}{4} \\
2 \sin 2 x & \text { if } & \frac{\pi}{4} < x \leq \frac{\pi}{2} \\
0 & \text { if } & x > \frac{\pi}{2}
\end{array}\right.
$$

Find the distribution function $F(x)$.
Solution. If $x \leq \pi / 4$, then $f(x)=0$, hence,

$$
F(x)=\int_{-\infty}^{x} f(x) d x=\int_{-\infty}^{x} 0 \cdot d x=0
$$

If $\pi / 4 < x \leq \pi / 2$, then

$$
F(x)=\int_{-\infty}^{\pi / 4} 0 \cdot d x+\int_{\pi / 4}^{x} 2 \sin 2 x d x=0-\left.\cos 2 x\right|_{\pi / 4} ^{x}=-\cos 2 x + \cos \frac{\pi}{2} = -\cos 2 x
$$

If $x > \pi / 2$, then

$$
F(x)=\int_{-\infty}^{\pi / 4} 0 \cdot d x+\int_{\pi / 4}^{\pi / 2} 2 \sin 2 x d x+\int_{\pi / 2}^{x} 0 \cdot d x=0-\left.\cos 2 x\right|_{\pi / 4} ^{\pi / 2}+0=1
$$

Thus, the random variable $X$ has the following distribution function:

$$
F(x)=\left\{\begin{array}{lll}
0 & \text { if } & x \leq \frac{\pi}{4} \\
-\cos 2 x & \text { if } & \frac{\pi}{4} < x \leq \frac{\pi}{2} \\
1 & \text { if } & x > \frac{\pi}{2}
\end{array}\right.
$$