Question

The following is the p.d.f. (probability density function) of a continuous random variable $X$:
$f\left(x\right)=\frac{x}{32},0<x<8=0$, otherwise
Find the expression for c.d.f. (cumulative distribution function) of $X$

• A. $\frac{x^2}{64}-1$
• B. $\frac{x^2}{64}$
• C. $\frac{x+1}{64}$
• D. $\frac{x^2-1}{64}$

Answer 1

The correct option is B $\frac{x^2}{64}$

c.d.f. of a continuous random variable $X$ is given by,

$f\left(x\right)={\int }_{-\infty }^{x}f\left(y\right)dy$

$\therefore f\left(x\right)={\int }_0^{x}f\left(y\right)dy={\int }_0^x\frac{y}{32}dy$

$={\left[\frac{y^2}{64}\right]}_0^x=\frac{x^2}{64}$

Thus, $F\left(x\right)=\frac{x^2}{64},x\in R$.