The shop can cut the cost of production per sandwich to $3.75. The revenue function would then be defined as R(x)=9x.
Step-by-step explanation:
We want to make as big profit as possible by making as little sandwiches as possible. This means that we want the profit, which is to be as big as possible.
For the first choice the profit per sandwich is dollars.
For the second choice the profit per sandwich is dollars.
For the third choice the profit per sandwich is dollars.
We see here that for the second choice the profit is greatest, therefore this choice is most suitably correct.
A random variable following a binomial distribution over trials with success probability has PMF
Because it's a proper probability distribution, you know that the sum of all the probabilities over the distribution's support must be 1, i.e.
The mean is given by the expected value of the distribution,
The remaining sum has a summand which is the PMF of yet another binomial distribution with trials and the same success probability, so the sum is 1 and you're left with
You can similarly derive the variance by computing , but I'll leave that as an exercise for you. You would find that , so the variance here would be
The standard deviation is just the square root of the variance, which is