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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<x<1
2−x, 1 ≤ x<2,
0, elsewhere.
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

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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

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