3.7 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

The probability density function (pdf) is one that goes (0,0) to (1,1) in a straight line, and drops back to zero (in a straight line) from 1 to 2.

<=120 hours means x<=1.2

(a) Area under the pdf from 0 to 1.2 equals

P (X<=1.2) = 0.5 (from 0 to1) + (1+0.8) / 2*0.2=0.5+.18=0.68

(b) area between 50 and 100 hours means 0.5<=x<=1

= (3/4) (0.5)

=0.375