00:01
For this exercise, we are told two things about a given random variable.
00:06
The time to failure, let's call it x, is exponentially distributed, and also that 10 % of components have failed by 1 ,000 hours.
00:18
So we can write that as the probability that x is no more than 10 ,000.
00:23
This should have another 0 in it.
00:25
Sorry, no, it's 1 ,000 hours.
00:27
So the probability that x is at most 1 ,000 hours, or the probability that the lifetime is at most 1 ,000 hours, is 10 % or 0 .1.
00:38
And then for a, we are asked to find the mean and standard deviation of the time to failure.
00:45
Now for an exponential random variable, the cdf or the cumulative distribution function is given by this formula.
01:12
It should be x not t.
01:18
And here, lambda is the rate parameter.
01:31
The mean is equal to 1 over lambda.
01:37
And the standard deviation is also equal to 1 over lamb.
01:44
So for this random variable, the mean is equal to 1 over lambda.
01:55
So we have to find lambda.
01:59
We know that the probability that x is at most 1 ,000 is equal to 0 .1.
02:14
So this means that 1 minus e to the negative lambda times 1 ,000 is equal to 0 .1.
02:39
And if we take the natural logarithm of both sides, and so for lambda, this gives us approximately 0 .00 ,000, and so continuing with this, the mean is equal to 1 over 0 .000, which is 10 ,000.
03:40
And this is in units of hours.
03:46
Now if i was rounding here, or rather i was rounding here, if you did it to more decimal places, you'd get a more accurate answer of 9 ,491 .22 hours.
04:05
So let's use this value for parts b and c.
04:24
For b we were asked for the probability that the component is still working after 5 ,000 hours.
04:30
This is the probability that x is greater than 5 ,000.
04:37
This can be re -expressed as 1 minus.
04:40
The probability that x is at most 5 ,000.
04:47
And then we can use the cdf and put 5 ,000 hours into the cdf to find this probability.
04:56
So this is equal to e to the negative 5 ,000 times 0 .000.
05:08
0 .001...