00:01
Okay, so we have a random variable x, which represents the lifetime of a certain type of electronic device measured in hours.
00:10
And we have the density f of x is given by this function.
00:14
So it's 10 over x squared when x is greater than are equal to 10 and 0 for x less than 10.
00:20
So we want to firstly find the probability that the device will fail in the first 15 hours.
00:27
So this means the lifetime of the device is less than 15.
00:31
So we're looking for the probability that x is less than or equal to 15.
00:37
So all we need to do is integrate the density, f of x, over an appropriate interval.
00:43
So we need to integrate from some value up to 15.
00:47
And this value is going to be 10 because the function is zero if x is less than 10.
00:52
So we need to evaluate the integral from 10 to 15 of f of x.
00:57
So now all we need to do is some integration, 10, 15.
01:01
And this is 10 over x squared d x so this is 10 times the integral from 10 to 15 of x to the minus 2 d x so if we integrate x to the minus 2 we get x to the minus 1 over minus 1 and then we need to evaluate at 15 and 10 so this is the same as minus 10 times 1 over x evaluated at 15 and 10 so plug in these in we get minus 10 times 1.
01:31
1 over 15 minus 1 over 10.
01:35
So let's put this over a common denominator.
01:38
Let's say a common denominator of 150.
01:42
So this would be 10 over, so this would be 10 over 150.
01:50
Yep, minus this would be 15 over 150.
01:56
This looks correct.
01:58
So if we subtract these, we get minus 10 times minus 5...