Question

The total number of hours, measured in units of 100 hours. that a family runs a vacuum cleaner over a period of one year is a continuous random variable X that has the density function
N: the number of

x, /2— if 0 < x < 1 f (x) =- x, if 1 < x < 1 0, otherwise.

Find the probability that over a period of one year, a family runs their vacuum cleaner
(a) less than 120 hours; (b) between 50 and 100 hours.

Answer 1

The total number of hours, measured in units of 100 hours. that a family runs a vacuum cleaner over a period of one year is a continuous random variable X that has the density function

f(x) = x ; 0 < x < 1

= 2 - x ; 1 x < 2

= 0, otherwise

(a) P( less than 120 hours) = 1- P(x > 120 hours) = 1 - P(x > 1.2)

= 1 -

= 1 - [2x - x²/2]²_1.2

= 1 - [2 * (2 - 1.2) - 1/2 * (2² -1.2²)] = 1 - 0.32 = 0.68

P(X < 120 hours) = 0.68

(b) P(50 hours < X < 100 hours) = P(0.5 < X < 1.0)

=

= [x²/2]¹_0.5

= 1/2 (1 - 0.5²) = 0.375

P(50 hours < X < 100 hours) = 0.375