The gap width between recording head and tape is an importan

The gap width between recording head and tape is an important property of a magnetic recording machine. The width, in µm, is a continuous random variable with probability density function

f(x) = 0.5x 0 < x < 2 µm

(a) what is the average gap width?

(b) what is the median gap width?

Solution

Give f(x) = (0.5) x                0<x<2

Mean = E(x) = x . f(x) dx

                     = x . (0.5) x dx

                      = 0.5 x ² dx

                      = 0.5 [x³ / 3]₀²

                      = 0.5(8/3)

     Mean = 1.3333 µm

in case of continuous distribution median is the point which divides the total area into two equal parts thus if M is the median then

₀^M f(x) dx = 1/2

₀^M     (0.5) x    dx = 1/2

0.5 ₀^M (x) dx = 1/2

(0.5) (x²/2)₀^M = 1/2

(0.5) M² = 1

M² = 1 / 0.5

M² = 2

Median = 1.414 µm