Application: Expectation and Variance in Business Decision-Making

The real payoff of everything built up in this chapter is that mathematical expectation and variance turn an uncertain business situation into two clear numbers a decision can be based on.

Suppose a trader is deciding whether to stock a perishable item for a festival sale. Demand is uncertain, so the profit $X$ earned is itself a random variable — it might be a healthy profit if demand is high, a modest profit if demand is moderate, or even a loss if most of the stock goes unsold and has to be discarded at a throwaway price. If the trader can estimate, from past experience, the probability of each of these demand scenarios, the profit figures and their probabilities together form a probability distribution exactly like the ones built earlier in this chapter.

Once that distribution is written down:

• $E(X)$, the expected profit, answers: "if this decision were repeated across many similar festival seasons, what would the average outcome be?" A positive expected profit is generally a first green light to proceed; comparing $E(X)$ across two or more competing proposals (say, two different products to stock) ranks them by their average payoff.
• $Var(X)$ or $SD(X)$, the risk, answers a different question: "how much could the actual result differ from that average?" A proposal with a lower expected profit but a much smaller standard deviation may still be the sensible choice for a risk-averse trader, while one with a higher expected profit but a very large spread might be rejected as too uncertain, or accepted only by someone able to absorb a possible loss. …