First, let me state the problem:
""Customers at Fred's cafe win a 100 dollar prize if their cash receipts show a star on each of the five consecutive days Monday...Friday in any one week. The cash register is programmed to print stars on a randomly select 10% of the receipts. If Mark eats at Fred's once each day for four consecutive weeks and the appearance of stars is an independent process, what is the standard deviation of X, where X is the number of dollars won by Mark in the four-week period""
The probability model that can best solve this would be a Binomial with parameters n=4 and p=0.15.
Folks can quickly solve this creating a random variable Y=100X and just getting the Variance of the binomial and taking the root.
The variance for Binomial is np(1−p)
Var[100X]=1002Var[X].
This method seems significantly more peaceful than finding the second moment and first moment of this new random variable Y.
My issue comes from this allowance of multiplying X by 100 for the expected payout for this particular problem. How is it that we translate this word problem in such a way? When n=0 this 100 shouldn't be getting multiplied by the probability I thought.
Can someone help me make this jump?
