Question

1.The pdf of a continuous random variable x is given by f(x)={c/x1/2 ,0<x<4 both inclusive

{0 ,elsewhere

find the value of the constant c,the cdf of x and p(x>or equal to 1)

Answer 1

$$\displaystyle\int_{-\infin}^{\infin}f(x)dx=\displaystyle\int_{0}^{4}\dfrac{c}{x^{1/2}}dx=c[2\sqrt{x}]\begin{matrix} 4 \\ 0 \end{matrix}=4c=1$$

$$c=\dfrac{1}{4}$$

$$f(x)=\begin{cases} \dfrac{1}{4\sqrt{x}} & 0\leq x\leq 4\\ 0 & elsewhere \end{cases}$$

$$F(x)=\displaystyle\int_{-\infin}^{x}f(x)dx, -\infin<x<\infin$$

$$F(x)=\begin{cases} 0, & x<0 \\ \dfrac{\sqrt{x}}{2} & 0\leq x< 4\\ 1 & x\geq4 \end{cases}$$

$$P(x\geq1)=1-F(1)=1- \dfrac{\sqrt{1}}{2}= \dfrac{1}{2}$$