Question

The recoded temperature in an engine is a r.v.X whose pdf is given by
f(x) = 
n(1 − x)
n−1
, : 0 < x < 1
0, : x < 0
where n ⩾ 1 is a known integer
(a) Show that f is indeed a pdf.
[3 Marks]
(b) Determine the corresponding cdf

Answer 1

(a)

$$\displaystyle\int_{-\infin}^{\infin}f(x)dx=\displaystyle\int_{0}^{1}n(1-x)^{n-1}dx$$

$$=[-(1-x)^n]\begin{matrix} 1 \\ 0 \end{matrix}=-(0-1)=1, n\geq1$$

Therefore $f(x)$ is indeed a pdf.

(b)

$$F(x)=\displaystyle\int_{-\infin}^xf(t)dt$$

$0<x<1$

$$F(x)=\displaystyle\int_{0}^xn(1-t)^{n-1}dt=[-(1-t)^n]\begin{matrix} x \\ 0 \end{matrix}$$

$$=1-(1-x)^n$$

$$F(x)=\begin{cases} 0 &x<0\\ 1-(1-x)^n & 0\leq x<1\\ 1 & x\geq 1 \end{cases}$$