00:01
Here we are given a random variable, let's call it x, which is the time it takes a technician to solve a problem, and we are told that it's uniformly distributed on 30 seconds to 14 minutes.
00:17
So that's half a minute to 14 minutes.
00:22
The first question is, what are a and b? so for a uniform random variable, by convention, a is the lower endpoint, and b is a the upper endpoint on the interval for which the probability is non -zero.
00:42
The probability density function for a continuous uniform random variable is given by this formula.
01:09
And so we have a is equal to 0 .5 and b is 14.
01:14
Now i've decided to use minutes but you could use seconds.
01:17
So we have it would be 30 seconds to 60 times 14 which would be 840 seconds.
01:24
But it just seems a bit more natural to use minutes.
01:27
But the main thing is both endpoints have to be in the same units.
01:32
We are asked for the mean.
01:35
So that's the mean time to resolve the problem.
01:39
For a uniform random variable, this is a plus b over 2, and this is 7 .25 minutes.
02:00
And then for a third question, we're asked, what is the standard deviation? and for a uniform random variable, the standard deviation is the square root of the following.
02:37
And this comes out to approximately 3 .90.
02:46
And then we're asked what percent take more than five minutes to resolve.
02:51
So this is the probability that x is greater than 5.
02:57
And this can be found by integrating or by finding the integral of the probability density function, which would be 1 over 13 and a half from 5 up to its maximum time of 14 minutes.
03:22
And this comes out to approximately 0 .67 or 66 .67%...