Question

the pdf of a continuous random variable x is given by f(x)=$\int c/$ square root x where 0 <x<4 find the constant c and find the cumulative distribution function of x then compute P(x>1)

Answer 1

On of the properties of pdf is: $\int_{-\infty}^{+\infty} f(x)dx = 1$, where f(x) is the pdf.

In the given case: $\int_{-\infty}^{+\infty} {\frac c {\sqrt x}}dx = 1\to\int_0^4 {\frac c {\sqrt x}}dx = 1\to 2c\sqrt{4}-2c\sqrt{0}=1\to$

$\to 4c=1\to c = 0.25$

The cdf is equal to 0 when x≤0, equal to 1 when x≥4, and equal to $\int_0^x {\frac 1 {4\sqrt t}}dt$ when x is 0<x<4

$\int_0^x {\frac 1 {4\sqrt t}}dt =0.25\int_0^x {\frac 1 {\sqrt t}}dt=0.25(2\sqrt{x}-2\sqrt{0})={\frac {\sqrt{x}} 2}$

$P(x>1) = 1-P(x≤1)=1-0.5=0.5$