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.

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

Answer 1

Given :   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

To find : Show that f(x)  is a p.d.f.  (probability density function )

Solution:

f(x)  = 6x(1 - x)    0 ≤ x  ≤ 1

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

if      $\int\limits^b_a {f(x)} \, dx = 1$

f(x)  = 6x(1 - x)  

f(x) = 6x  - 6x²

$\int\limits^1_0 {(6x - 6x^2)} \, dx$

=  $[ \frac{6x^2}{2} - \frac{6x^3}{3} ] ^1_0$

= $[ 3x^2 - 2x^3 ] ^1_0$

= 3(1)² - 2(1)³  - ( 0 - 0)

= 3 - 2 - 0

= 1

$\int\limits^1_0 {(6x - 6x^2)} \, dx$  =  1

Hence f(x) = 6x(1 - x)    0 ≤ x  ≤ 1  is pdf  ( probability density function )

Learn more:

A random variable X has the probability distribution - Brainly.in

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