The following is the p.d.f. (ProbabiIity Density Function) of a continuous random variable X :
`f(x)=x/32,0<x<8`
= 0 otherwise
(a) Find the expression for c.d.f. (Cumulative Distribution Function) of X.
(b) Also find its value at x = 0.5 and 9.
Solution 1
(a) c.d.f. of a continuous random variable X is given by
`F(x)=int_(-oo)^xf(y)dy`
In the given density function f(x), range of X starts at '0'.
`therefore F(x)=int_0^xf(y)dy=int_0^xy/32dy=[y^2/64]^2=x^2/64`
`"Thus ",F(x)=x^2/64, AA x in R`
(b) `F(0.5) =(0.5)^2/64=1/256`
For any value of x ≥ 8, F(x)=1
∴ F(9)=1
Solution 2
Given p.d.f
`f(x)=x/32,0<x<8`
`c.d.f. = F(X)=int_0^xf(x)dx`
`=int_0^x x/32dx`
`=[x^2/64]_0^x`
`=X^2/64`
`"at "x = 0.5`
`F(X)=F(0.5)=(0.5)^2/64=0.25/64=0.0039`
For any value of x greater than 8
`F(X)=1`
`therefore F(9)=1`
