Distributions and Expected Value

An interviewer asks, "Our website records a few purchases per minute. Which probability distribution would you use to model the number of purchases per minute? How would your choice change if you were modeling the time until the next purchase?"

An incorrect approach assumes a normal distribution because the data are approximately normal. This approach selects a distribution without examining the characteristics of the data or the assumptions of the model. Purchase counts are nonnegative integers. When the average count is low, their distribution may be right-skewed rather than approximately normal. A more effective approach starts by identifying the variable being modeled. A Poisson distribution may model the number of purchases in a fixed interval. In contrast, an exponential distribution may model the waiting time between purchases, assuming events follow a homogeneous Poisson process. The following section explains how to select a distribution based on the variable and its underlying assumptions.

Apply the same reasoning to distinguish questions that require different probability distributions. The number of purchases per minute represents an event count within a fixed time interval. The time until the next purchase represents a continuous waiting-time variable. The number of conversions among 10,000 emails represents a count of successes across a fixed number of trials. Temperature readings centered around 22 degrees represent continuous numerical measurements. Although all four scenarios involve uncertainty, the type of variable and the assumptions of the underlying process determine which probability distributions are appropriate.

The table weighs the four common choices against the structure of the observation:

This table shows why distribution choice starts with the data-generating process rather than a formula we happen to remember.

Each probability distribution relies on assumptions that candidates should be prepared to explain and validate. A binomial model assumes a fixed number of independent trials, each with the same probability of conversion. A homogeneous Poisson process assumes a constant average event rate and independent event counts across nonoverlapping time intervals. The waiting time between events in a homogeneous Poisson process follows an exponential distribution with a constant event rate. A normal model assumes that the variable follows a continuous, symmetric, bell-shaped distribution characterized by its mean and variance.

A low purchase rate (purchases per minute) alone does not establish that a Poisson distribution is appropriate. Flash sales, promotional campaigns, and daily demand patterns can cause purchase arrivals to cluster or the event rate to vary over time. A well-supported response would be: “I'd consider a Poisson model if purchases occur independently at an approximately constant average rate over the observation period. I'd validate these assumptions before using the model.”

Expected profit makes the distribution useful

Once the outcome structure is clear, expected value is the long-run average result from repeating the same uncertain decision under the same assumptions. It tells us whether an email send has positive expected profit. It does not guarantee that every individual send earns money.

An interviewer might ask, “An email has a 2% conversion probability. Each conversion produces 50 dollars in revenue, and each send costs 0.75 dollars. What is the expected profit per send?”

A weak answer says, “A conversion earns 50 dollars, so the campaign makes money.” That answer ignores the 98% of sends that do not convert and ignores the sending cost. The error is not arithmetic. The candidate has not weighted each possible outcome by its probability.

A strong answer defines profit per send as $X$ ...