00:01
The scenario here is that we have a carton that is packed with 10 boxes, and that's how it ships.
00:07
The weights of boxes, let's call that random variable x, are normally distributed with a mean weight of 500 grams, and a standard deviation of 10 grams, and the weights of the carton are normally distributed with the mean of 100 grams, and a standard deviation of 12 grams.
00:25
So for part a, we're asked to calculate the mean and variance for a full carton, which contains 10 boxes, so first we have the carton itself.
00:38
So let's call the total weight w.
00:42
We have the carton itself, and inside the carton are 10 boxes.
00:56
So we have one draw from this distribution, 10 draws from this distribution, and we're putting them all together.
01:05
Now if we want to know the mean of w, it's equal to the sum of the means of all of these individual components.
01:15
So that's the mean of y plus 10 times the mean of x.
01:26
So we have 100 plus 10 times 500, and this comes out to 50 ,100.
01:39
So the mean of the total package is 5 ,100 grams.
01:45
Now the variance of w is equal to the sum of the variances of the individual components that make it up.
01:54
So we have this is equal to the variance of y plus 10 times the variance of the variance.
02:00
Variance of x.
02:15
And this comes out to 1 ,144...