Farmers’ market A farmer has 100 lb of apples and 50 lb of...

Farmers’ market A farmer has 100 lb of apples and 50 lb of potatoes for sale. The market price for apples (per pound) each day is a random variable with a mean of 0.5 dollars and a standard deviation of 0.2 dollars. Similarly, for a pound of potatoes, the mean price is 0.3 dollars and the standard deviation is 0.1 dollars. It also costs him 2 dollars to bring all the apples and potatoes to the market. The market is busy with eager shoppers, so we can assume that he’ll be able to sell all of each type of produce at that day’s price. a) Define your random variables, and use them to express the farmer’s net income. b) Find the mean. c) Find the standard deviation of the net income. d) Do you need to make any assumptions in calculating the mean? How about the standard deviation?

Farmers’ market A farmer has 100 lb of apples and 50 lb of potatoes for sale. The market price for apples (per pound) each day is a random variable with a mean of 0.5 dollars and a standard deviation of 0.2 dollars. Similarly, for a pound of potatoes, the mean price is 0.3 dollars and the standard deviation is 0.1 dollars. It also costs him 2 dollars to bring all the apples and potatoes to the market. The market is busy with eager shoppers, so we can assume that he’ll be able to sell all of each type of produce at that day’s price.
a) Define your random variables, and use them to express the farmer’s net income.
b) Find the mean.
c) Find the standard deviation of the net income.
d) Do you need to make any assumptions in calculating the mean? How about the standard deviation?

Key Concepts

Linear Transformation of Random Variables

A linear transformation involves scaling and/or shifting a random variable by applying operations such as multiplication by a constant or addition of a constant. This affects both the expected value and the variance in predictable ways, which is vital when calculating derived quantities such as total income from selling weighted amounts of goods.

Random Variables

Random variables are functions that assign a numerical value to each outcome of a random process. In the context of uncertain market prices, they model the unpredictable fluctuations in the price per unit of goods. This concept is essential for quantifying and managing uncertainty in economic scenarios.

Expected Value

The expected value represents the long-run average or mean of a random variable if an experiment is repeated many times. It is found by weighting each outcome by its probability and summing the products. In applications such as calculating net income, the expected value provides insight into the average financial return under uncertainty.

Variance and Standard Deviation

Variance measures the dispersion of a random variable's values around its expected value, indicating the degree of spread or risk. The standard deviation, the square root of the variance, provides this measure in the same units as the variable, making it easier to interpret. These concepts are crucial in assessing the risk associated with uncertain outcomes like fluctuating prices.

Independence Assumption

Independence assumes that the random variables involved do not influence each other, meaning the occurrence of one does not affect the probability distribution of the other. This assumption simplifies calculations of combined variances, as the variance of the sum of independent variables is the sum of their variances. It is particularly important when assessing risk in multi-variable economic models.

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