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The density function of the continuous random variable x, the total number of hours, in units of 100 hours, that a family runs a vacuum cleaner over a period of one year is given below: f(x) = { x, 0 < x < 1 8 − x, 1 ≤ x < 2 0, otherwise Find the average number of hours per year that families run their vacuum cleaners?

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Answer:

1000 hours

Step-by-step explanation:

∫xf(x)dx = ∫x*xdx (1,0) + ∫x*(8-x)dx (2,1)

∫xf(x)dx = ∫x²dx (1,0) + ∫8x - x²dx (2,1)

∫xf(x)dx = ∫x²dx (1,0) + ∫8xdx (2,1) - ∫x²dx (2,1)

∫xf(x)dx = x³/3(1,0) + 8x²/2 (2,1) - x³/3(2,1)

∫xf(x)dx = 1/3 + 8(4/2 - 1/2) - (8/3 - 1/3)

∫xf(x)dx = 1/3 + 8(2-0.5) - (7/3)

∫xf(x)dx = 1/3 + 8(1.5) - 7/3

∫xf(x)dx = 1/3 + 12 - 7/3

∫xf(x)dx = - 6/3 + 12

∫xf(x)dx = - 2 + 12

∫xf(x)dx = 10

The average number of hours the family runs their vacuum cleaner in units of 100 hours

100 * 10 hours = 1000 hours

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