<s>[INST] Where Is my wrong The random variables given for the PDF distribution function:
$$
p_X(x) = 
\begin{cases} 
    \frac{a}{7}x, & 2 <x\leq 3   \\
    0, & x\notin(2,3] \\
\end{cases}
$$
$a)$ Find value of parametric $a$, and $F_X(x)$ (CDF).
$b)$ find the probability that the random variables is part of interval $(-\frac{5}{2},\frac{5}{2})\$
My attemp is: First part
$a)$ $$\int_2^3\frac{a}{7}xdx=1\Rightarrow a=\frac{14}{5}$$
Now we have:
$$
F_X(x) = 
\begin{cases} 
    \frac{14}{35}x, & 2 <x\leq 3   \\
    0, & x\notin(2,3] \\
\end{cases}
$$
Case 1: If $x<2$ we have $F_X(x)=0$
Case 2: If $2<x\leq 3$ we have: $F_X(x)=\int_2^x\frac{14}{35}udu\Rightarrow F_X(x)=\frac{7}{35}x^2-\frac{28}{35}$
Case 3: If $x>3$ we have $F_X(x)=1$
Now for the second part
$b)$ If $x<2$ for the function $F_X(x)$ we have $F_X(x)=0$ so the values ​​of the interval remain $(\2,\frac{5}{2})\$. From here
$$P(X\in(2,\frac{5}{2}))=F(\frac{5}{2})-F(2)=\frac{7}{35}\frac{25}{4}-\frac{28}{35}-\frac{7}{35}4-\frac{28}{35}=-\frac{161}{120}????$$
i don't know where i went wrong. Please help me. Thanks
 [/INST] You're wrong at the last line by substracting $\frac{28}{35}$ twice, since you forgot to flip the sign the second time.
$$F_X(\frac{5}{2}) - F_X(2) = \frac{1}{5}\frac{25}{4} - \frac{4}{5} - \big(\frac{1}{5}4 - \frac{4}{5}\big) = \frac{1}{5}\big(\frac{25}{4} - 4\big)$$
</s>