A bottling machine is supposed to fill containers with 16 o

- A bottling machine is supposed to fill containers with 16 ounces of liquid. The number of ounces X by which the machine varies from the target value of 16 is a random variable with probability density function: f(x)= 1+x, x[-1,0] and 1-x, x(0,1] and 0, otherwise. Find the probability that the machine puts in less than 15.75 ounces of liquid in a bottle.

Solution

f ( x ) = 1 + x when -1 <= x <= 0 ....
   = 1 - x when 0 < x <=1 ....
   = 0 otherwise

E ( X) = Integration from -1 to 0 [ x*( 1+x) dx ] + integration from 0 to 1 [ x*( 1 - x) dx ]
   = 0 ...

E ( X^2) = Integration from -1 to 0 [ x^2*( 1+x) dx ] + integration from 0 to 1 [ x^2*( 1 - x) dx ]
   = 1 /6...

var ( x) = 1 /6..so, s.d ( x) = sqrt ( 1 / 6 )

P [ X < 15.75 ] = P [ Z < ( 15.75 - 16 ) / sqrt (1/6) ] = P [ Z < -0.6123724 ] = 0.2701457....