# Expected Monthly Profit for a Small Manufacturing Firm

• Of Mike and Men
In summary: Is your official answer with a different (e.g. 0.3, 0.2) set of probabilities?In summary, using the probabilities given, the expected monthly profit for a small manufacturing firm selling machines is $3,800. This is calculated by taking the sum of the profit for each possible outcome (1 machine sold, 2 machines sold) multiplied by their respective probabilities and adding the mean of X. The mean of X is calculated by multiplying the number of machines sold by their respective probabilities and then adding them together. The variance can also be used to find the expected profit by calculating the mean of X squared. Of Mike and Men ## Homework Statement A small manufacturing firm sells 1 machine per month with 0.3 probability; it sells 2 machines per month with 0.1 probability; it never sells more than 2 machines per month. If X represents the number of machines sold per month and the monthly profit is 2X2 + 3X + 1 (in thousands of dollars), find the expected monthly profit. ## Homework Equations ## The Attempt at a Solution E(2X2 + 3X + 1) = ∑(2X2 + 3X + 1)f(x), x = 1, 2 = (2+3+1)(.3) + (8+6+1)(.1) = 1.8 + 1.5 = 3.3 3.3 * 1000 =$3,300

The answer in the back is $3,800. If I take my 3.3 and add the mean of E(X) = .3(1) + 2(.1) = 0.5, I get 3.8 * 1000 =$3,800. I'm not sure if this is coincidence or actually how you solve the problem. If it's how you solve the problem, I don't understand why you add the mean of X. If it's not the way you solve it, I'm not sure what to do and would like some hints (but not a solution -- if possible) as to where to go.

Thanks.

You are missing one possible outcome ...

Edit: Not that that actually fixes the problem ...

Of Mike and Men said:

## Homework Statement

A small manufacturing firm sells 1 machine per month with 0.3 probability; it sells 2 machines per month with 0.1 probability; it never sells more than 2 machines per month. If X represents the number of machines sold per month and the monthly profit is 2X2 + 3X + 1 (in thousands of dollars), find the expected monthly profit.

## The Attempt at a Solution

E(2X2 + 3X + 1) = ∑(2X2 + 3X + 1)f(x), x = 1, 2
...the mean of E(X) = .3(1) + 2(.1) = 0.5

So profit ##= 2X^2 + 3X + 1##

##E[profit] = E[2X^2 + 3X + 1] = E[2X^2] + E[3X] + E[1]##

by linearity of expectations.

This last line should be simplified a bit -- how would you do it? You correctly calculated ##E[X]##. What is ##E[X^2]##? Or if you prefer, what is the Variance -- you can recover ##E[X^2]## from that.

I'm getting an answer a touch higher than the 3.8 (thousand) mentioned as the official one, though.

## 1. What is the definition of "Expected Monthly Profit"?

The expected monthly profit is an estimate of the amount of money a company or individual expects to earn in a given month. It takes into account factors such as revenue, expenses, and market trends to predict the potential profitability of a business.

## 2. How is "Expected Monthly Profit" calculated?

The expected monthly profit is typically calculated by subtracting the expected expenses from the expected revenue for a given month. This calculation can also take into account other factors such as projected sales growth or changes in market conditions.

## 3. Why is "Expected Monthly Profit" important for a business?

Expected monthly profit is important for a business because it helps to forecast the financial health of the company and make informed decisions about budgeting, investments, and growth strategies. It also serves as a benchmark for measuring actual performance and identifying areas for improvement.

## 4. Can "Expected Monthly Profit" be accurate?

While expected monthly profit is an estimate and may not always be exact, it can be a useful tool for financial planning and decision making. The accuracy of the estimate depends on the quality of the data and assumptions used in the calculation.

## 5. How can a business increase its "Expected Monthly Profit"?

There are several ways a business can increase its expected monthly profit, including reducing expenses, improving efficiency, increasing sales, and expanding into new markets. It is important to regularly review and adjust the expected monthly profit to reflect changes in the business environment and make necessary adjustments to reach profitability goals.

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