Discrete Random Variables

Research Focus Room
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Hope you guys had a good endsem and some well needed rest.

Last time, we explored random variables and key properties such as expectation, variance, and correlation. This week, let’s dive into several commonly occurring discrete random variables, the ones you’ll frequently encounter both in the real world. Developing intuition for these distributions helps you quickly recognize patterns in new problems and choose the right analytical tools.

Common Discrete Distributions

Here are some of the most widely used discrete distributions:

  1. Bernoulli Distribution
  2. Binomial Distribution
  3. Uniform (Discrete) Distribution
  4. Geometric Distribution
  5. Poisson Distribution
  6. Hypergeometric Distribution

Take some time to read about these distributions and their typical real-world applications, as well as try out the MCQs on bequant.dev, before attempting the following:

Modeling Customer Arrivals at an Ice-Cream Shop

You run a small ice-cream shop and want to model how many customers show up during a given 10-minute window. Depending on how your business operates and what constraints exist, different discrete distributions are appropriate.

Here are a few simple scenarios:

Scenario A — Lots of potential customers passing by randomly

Your shop is on a busy street. Many potential passersby walk by, each with a very small and roughly independent chance of entering the shop. Historical data shows that on average 6 customers arrive during any 10-minute period.

Scenario B — A fixed number of regular customers

Your shop is on a busy street. Many potential passersby walk by, each with a very small and roughly independent chance of entering the shop. Historical data shows that on average 6 customers arrive during any 10-minute period.

Scenario C — Fixed customers, but limited seating capacity

Similar to Scenario B, but your shop can serve only up to 8 customers in 10 minutes. After that, additional customers are turned away.

Your Tasks

  1. Identify which scenario corresponds to which distribution and explain why.
  2. Compute the expected number and variance of customers per 10 mins each scenario.
  3. Explain why choosing the wrong distribution might give bad predictions about staffing or inventory.
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