00:01
In problem 17, given the dynasty function for the random variable y, which represents the time required to complete a one hour exam.
00:10
For party a, we want to find c.
00:12
To make this function, a dynasty function, it should have the property that the integration for f of y, dy, along its interval, from 0 to 1, must be equal to 1.
00:29
Then integration from 0 to 1 4 f of y c squared c multiplied by a squared plus y d y equals 1 then let's integrate the first term integration of i squared is y cubed divided by 3 then it's c y cubed divided by 3 plus integration of y is y squared is y squared divided by 2 and the limits of the integration is from 0 to 1 this equals 1 let's substitute first by the upper limit then we have c divided by 3 plus 1 half minus we substitute by the lower limit which gives 0 equals 1 then c equals 1 minus half multiplied by 3 which is 3 half this value makes the density function a density function for probability for part b we want to find f of y if y is by definition is the integration along the interval from minus infinity of y it's not along the interval if of y is integration from minus infinity to y for f of y d y this makes us see the values for f of y for the value of y from minus infinity to infinity we can see that we have from minus infinity to zero it's zero and from zero to one is given by this function and from infinity to zero sorry from one to infinity it's zero again then we have three intervals and two of them has a has the value of the function zero and then we will follow on this interval now.
02:48
So get this integration, it equals the integration from minus infinity to 0 for f of y, d .y plus the integration from 0 to y, f of y, d y, then we have this value equal 0 because the function is defined as 0 from minus infinity to 0 and here we can write the expression in our problem.
03:21
It equals the integration from 0 to y for c y squared.
03:26
C now is 3 halves, 3 halves, y squared plus y, d y.
03:37
Then its integration is 3 y cube divided by 3 divided by 2, then it's half y cube plus y squared divided by 2...