112. A random variable $X$ is subject to Simpson's law (the law of the isosceles triangle) on the interval $x \in[-c ; c]$ (Fig. 18). Find:

a) the probability density function of this random variable;

b) the probability of the variable $X$ falling into the interval $(c / 2, c)$.
The area bounded by the graph of the function $f(x)$, which characterizes the probability density function of a random variable $X$, and the $O x$ axis is equal to one. Since the graph of the probability density function of the random variable $X$ is shown in Fig. 18, the area of the triangle $A B C$ must be equal to one. Using this condition, we will find the analytical expression for the function $f(x)$.

![](https://cdn.mathpix.com/cropped/2024_05_22_c311c9dd990dc55c716eg-139.jpg?height=290&width=628&top_left_y=330&top_left_x=390)

Fig. 18

The area of $\triangle A B C$ is $S=\frac{1}{2} A C \cdot O B$; from this we get $O B=\frac{2 \cdot S}{A C}$. Since $S=1, A C=2 c$, then $O B=1 / c$.

The differential function of the distribution of the random variable $X$ for $x \in[-c ; c]$ is represented by two segments of straight lines. We will find the equations of these lines. We will look for each equation in the form of the equation of a straight line in segments on the axes:

$$
\frac{x}{a}+\frac{y}{b}=1
$$

where $a$ and $b$ are the segments cut off by the line on the $O x$ axis and the $O y$ axis, respectively; $x, y$ are the current coordinates.

For $-c \leq x \leq c$, the differential function of the random variable $X$ is 0.

Combining the obtained results, we will describe the probability density function of the random variable $X$ over the entire number line:

$$
f(x)=\left\{\begin{array}{ccc}
0 & \text { if } & x < -c \\
\frac{1}{c}\left(1+\frac{x}{c}\right) & \text { if } & -c \leq x \leq 0 \\
\frac{1}{c}\left(1-\frac{x}{c}\right) & \text { if } & 0 < x \leq c \\
0 & \text { if } & x > c
\end{array}\right.
$$

The last expression can be written better:

$$
f(x)=\left\{\begin{array}{ccc}
\frac{1}{c}\left(1-\frac{|x|}{c}\right) & \text { if } & -c \leq x \leq c \\
0 & \text { if } & x < -c \text { or } x > c
\end{array}\right.
$$

b) The probability of the random variable $X$ falling into the interval $(-c / 2 ; c)$ can be found using formula (52). In this interval, for $x \in(-c / 2 ; 0)$ and for $x \in(0 ; c)$, the function $f(x)$ is described by two different equations, so the desired probability is

$$
P\left(-\frac{c}{2}<X<c\right)=\int_{-c / 2}^{0} \frac{1}{c}\left(1+\frac{x}{c}\right) d x+\int_{0}^{c} \frac{1}{c}\left(1-\frac{x}{c}\right) d x=\frac{7}{8}
$$