00:02
In the asked question, we have the dynasty function for the random variable x, where f of x equals x divided by 16 for x between 0 and 4 and 5 minus x when x is between 4 and 5 and 5 elsewhere, where x is a total number of hours in units of 100 hours.
00:36
That the family runs a vacuum cleaner over a period of one year then x here has a unit of hours in units of 100 hours then it's in 100 hours suppose that x equals 3 for example then this means the family runs 300 hours of a vacuum cleaner over a period of one year for part a it's required to find the average number of hours per year that families run their vacuum cleaners in hours, not in 100 hours, in hours.
01:19
To do so, the average number is expected value of the random variable.
01:26
It simply equals the integral from minus infinity to infinity for f of x d x.
01:36
F of x is defined 0 elsewhere and it's defined as x by 16 between 0 and 4, then 5, from 0 to 4, it's x divided by 16 d x plus from 4 to 5 it's 5 minus x the x and of course plus 0.
01:59
The integral of x is x squared divided by 2 and we have 16 then it's x squared divided by 32.
02:07
We substitute from 0 to 4 plus 5 minus x the integral is 5x minus x squared divided by 2.
02:15
You substitute from 4 to 5 sorry i've made a mistake this integral should give us 1 the expected value of x is the integral of x multiplied by f of x i'm sorry for this mistake then we should integrate x squared and here we should have x multiplied by 5 minus x and i should remove all these let's integrate x squared divided by 16 x squared has an integral of x cubed divided by 3 and we have 16 in the denominator then 16 by 3 it's 48 we substitute from 0 to 4 plus x multiplied by 5 is 5 x its integral is 5x its integral is 5 x squared divided by 2 minus x multiplied by x is x squared its integral is x cubed divided by 3 from 4 to 5 it equals 1 over 48 multiplied by we substitute by x equals 4 first then it's 4 cubed minus x equals 0 gives 0 plus we substitute by x equals 5 first it's 5 multiplied by 5 squared it's 125 divided by 2 minus 125 divided by 3 minus we substitute by x equals 4 it is 5 multiplied by 16 it's 80 divided by 2 minus 4 cubed which is 64 divided by 3 when we get this it equals 3 .5 300 hours then it's 3 .500 hours we transform it by multiplying it to 100 in 100 hours we transform it by multiplying it to 100 in 100 hours.
04:37
Hours as a unit, we multiply by 100, then it's 350 hours.
04:45
This is the expected value.
04:50
For part b, it's required to find the standard deviation of the number of hours period that families run their vacuum cleaners...