Question

The diameter of an electric cable, Say X, is assumed to be continuous random variable with p.d.f f(x)=6x(1-x), 0<x<1.

A. Show that f(x) is a p.d.f.

B. Determine a number b such that P(X<b)= P(X >b).

C. Find the mean of x.

Answer 1

A.

Function f(x) is a probability density function in range a to b if

$\int^b_af(x)dx=1$

Then:

$\int^1_0(6x(1-x))dx=(3x^2-2x^3)|^1_0=3-2=1$

C.

$E(X)=\int^b_axf(x)dx=\int^1_0x(6x(1-x))dx=(2x^3-1.5x^4)|^1_0=2-1.5=0.5$

B.

$P(x<b)=\int^b_0f(x)dx$

$P(x>b)=1-\int^b_0f(x)dx$

Then:

$\int^b_0f(x)dx=1-\int^b_0f(x)dx$

$2\int^b_0f(x)dx=2\int^b_0(6x(1-x))dx=2(3x^2-2x^3)|^b_0=6b^2-4b^3=1$

$b=0.5$