Probability Distributions

Introduction

Business decisions are made under uncertainty. A retailer cannot know exactly how many customers will arrive tomorrow, a bank cannot know which applicants will default, and a service manager cannot know the precise time until the next customer request. Probability distributions provide a structured way to represent such uncertainty.

A probability distribution describes the possible values of a random variable and the probability associated with each value. In business analytics, distributions support demand forecasting, inventory planning, quality control, queue management, credit risk assessment, insurance pricing, and simulation.

This chapter focuses on four important distributions:

• the binomial distribution for a fixed number of independent yes/no outcomes;
• the Poisson distribution for counts of events in a fixed interval;
• the normal distribution for continuous measurements clustered around a mean;
• the exponential distribution for waiting times between events.

The correct choice depends on the nature of the variable, the measurement scale, and the assumptions underlying the model.

Learning Objectives

By the end of this chapter, you should be able to:

1. distinguish discrete and continuous random variables;
2. explain probability mass functions, density functions, means and variances;
3. calculate and interpret binomial, Poisson, normal and exponential probabilities;
4. select a suitable distribution for a business problem;
5. identify assumptions, limitations and risks of inappropriate modelling;
6. use probability distributions to support managerial decisions.

1. Random Variables and Distributions

A random variable assigns a numerical value to an uncertain outcome. A discrete random variable has countable values, such as the number of defective items in a batch. A continuous random variable can take any value within an interval, such as delivery time or product weight.

A discrete distribution is described by a probability mass function (PMF):

where X is the random variable, x is a possible value, and p(x) is its probability. A valid PMF satisfies:

For a continuous random variable, probabilities are described by a probability density function (PDF), f(x). The probability of an interval is the area under the density curve:

For a continuous variable, the probability of one exact point is zero: P(X = x) = 0. This does not mean that the value cannot occur; it means that probability is assigned to intervals rather than isolated points.

The cumulative distribution function (CDF) is applicable to both types of variable:

The expected value or mean represents the long-run average:

For a continuous variable, the corresponding expression is E(X) = ∫ x f(x) dx. The variance measures dispersion around the mean:

where μ = E(X). The standard deviation is σ = √Var(X), expressed in the same units as X.

A distribution is a model, not a guarantee. It approximates a real process under stated assumptions.

2. The Binomial Distribution

2.1 Conditions

The binomial distribution models the number of successes in n trials when:

1. there is a fixed number n of trials;
2. each trial has exactly two outcomes, conventionally success or failure;
3. the probability of success p is constant;
4. trials are independent;
5. X counts the number of successes.

If any item is seriously violated, a binomial model may be unsuitable. For example, sampling without replacement from a small population usually creates dependence and may require a hypergeometric model.

2.2 Probability formula

If X ∼ Binomial(n,p), then the probability of exactly x successes is:

where C(n,x) = n! ÷ [x!(n−x)!], n is the number of trials, x = 0, 1, ..., n, and p is the success probability. The complement 1 − p is the failure probability.

The mean and variance are:

The standard deviation is √[np(1−p). These formulas count successes, so the units are numbers of successes or trials.

Worked Example 1: Sales conversion

A digital campaign sends offers to 20 customers. Historical evidence suggests that each customer has a 0.10 probability of making a purchase, independently of the others. Let X be the number of purchases.

The probability of exactly three purchases is:

Since C(20,3) = 1140,

Thus, the probability is approximately 19.0%. The expected number of purchases is np = 20 × 0.10 = 2, and the variance is 20 × 0.10 × 0.90 = 1.8, giving a standard deviation of approximately 1.342 purchases.

For “at least one” purchase, it is usually easier to use the complement:

Therefore, there is approximately an 87.8% chance of at least one purchase.

Business uses

Binomial models are useful for conversion counts, loan approvals, defective items in a fixed sample, employee training passes, and customer responses to a promotion. Analysts should investigate whether observations are genuinely independent and whether p remains stable across customers or time periods.

3. The Poisson Distribution

3.1 Purpose and assumptions

The Poisson distribution models the number of events occurring in a fixed interval of time, distance, area or volume. Examples include customer arrivals per hour, network failures per day, insurance claims per month, or defects per metre of material.

The standard assumptions are:

• events occur independently;
• the average event rate is constant over the interval;
• events occur singly rather than in simultaneous clusters;
• the probability of an event in a very small interval is proportional to the interval length.

If X is the count in an interval and X ∼ Poisson(λ), λ is the expected number of events in that interval. The probability of x events is:

Here, e ≈ 2.71828. The mean and variance are both λ:

The rate must be matched to the interval. If a call centre averages 12 calls per hour, the expected number in 15 minutes is λ = 12 × 0.25 = 3.

Worked Example 2: Support requests

A technical support team receives an average of 4 urgent requests per hour. Assuming a constant rate and independent arrivals, the probability of exactly two requests in the next hour is:

The probability of no requests is P(X=0)=e⁻⁴≈0.0183. Hence, the probability of at least one request is approximately 98.17%.

If management wants the probability of at most two requests, then:

Poisson and binomial relationship

The Poisson distribution can approximate a binomial distribution when n is large, p is small, and λ = np. For example, rare claims among a very large number of policyholders may be modelled this way. This is an approximation, not an automatic replacement.

4. The Normal Distribution

4.1 Characteristics

The normal distribution is continuous, bell-shaped and symmetric about its mean μ. It is determined by two parameters: μ and standard deviation σ, where σ > 0. Its density is:

The value of f(x) is a density, not the probability at exactly x. Probabilities are areas under the curve. For a normal variable, mean, median and mode are equal.

The empirical rule provides a useful approximation:

• approximately 68% of observations lie within μ ± σ;
• approximately 95% lie within μ ± 2σ;
• approximately 99.7% lie within μ ± 3σ.

These percentages apply to a normal distribution and are approximations.

4.2 Standardisation

To calculate probabilities using a standard normal table or software, convert X to a z-score:

Z measures how many standard deviations an observation lies above or below the mean. The standard normal variable has mean 0 and standard deviation 1.

Worked Example 3: Delivery times

Suppose delivery time X is approximately normal with mean 40 minutes and standard deviation 8 minutes. The probability that a delivery takes less than 52 minutes is found by standardising:

From a standard normal table, P(Z ≤ 1.5) ≈ 0.9332. Thus, approximately 93.32% of deliveries take less than 52 minutes.

The probability of a delivery between 32 and 48 minutes is:

Therefore, P(32 ≤ X ≤ 48) = P(−1 ≤ Z ≤ 1) ≈ 0.6827.

Applications and cautions

Normal models are often used for measurement errors, aggregated demand, process characteristics and approximately symmetric financial or operational measures. They can also arise through the central limit theorem: under suitable conditions, the distribution of a sample mean becomes approximately normal as sample size increases.

However, revenue, claim sizes and waiting times may be skewed and should not automatically be treated as normal. A normal model also permits negative values, which may be impossible for variables such as demand or processing time. Analysts should inspect histograms, quantile plots and business constraints before applying it.

5. The Exponential Distribution

5.1 Waiting-time interpretation

The exponential distribution models the waiting time between events in a Poisson process. If events occur at a constant rate λ per unit of time, the waiting time T to the next event has density:

The CDF and survival probability are:

The mean and variance are:

If λ is measured in calls per hour, the mean waiting time is measured in hours. The exponential distribution has the memoryless property:

This means that, under the model, the elapsed waiting time does not alter the distribution of the additional waiting time.

Worked Example 4: ATM arrivals

Customers arrive at an ATM at an average rate of 6 per hour. The average time between arrivals is 1÷6 hour, or 10 minutes. The probability that the next customer arrives within 5 minutes is obtained by expressing time in hours: t = 5÷60 = 1÷12. Therefore:

There is approximately a 39.35% chance that the next customer arrives within five minutes.

Distinguishing Poisson and exponential models

The Poisson distribution models a count during an interval. The exponential distribution models a waiting time until an event. They are linked, but they answer different questions.

6. Selecting an Appropriate Distribution

A practical selection process begins with the variable and the business mechanism rather than with a preferred formula.

The analyst should also ask:

1. Is the variable discrete or continuous?
2. Is the observation interval fixed?
3. Are outcomes independent?
4. Is the event rate constant?
5. Is the distribution strongly skewed?
6. Are there limits, clusters, seasonal effects or changing customer segments?

Parameter estimation is normally based on historical data. For a Poisson process, the sample mean is a natural estimate of λ. For a normal model, the sample mean and sample standard deviation estimate μ and σ. Estimates should be accompanied by data-quality checks and, where decisions are consequential, uncertainty intervals.

7. Integrated Case: Capacity Planning for an Online Support Centre

Background

This illustrative case concerns a subscription software company. The support centre receives urgent requests and wants to plan staffing for the next hour.

Problem

Historical data show an average of 8 urgent requests per hour during weekday afternoons. Managers need to estimate the probability of unusually high demand and the likely waiting time between requests.

Analysis

Let X be the number of urgent requests in one hour. A Poisson model is initially reasonable because X is a count in a fixed time interval. With λ = 8:

The probability of at least 10 requests is:

Thus, demand reaches 10 or more requests in approximately 28.34% of comparable hours. The mean waiting time between requests, under the corresponding exponential model, is:

Approach and solution

Management could schedule additional coverage when the forecast rate is high and use the tail probability as a workload indicator. However, the model should be tested by comparing observed counts with Poisson predictions. If requests arrive in bursts because incidents affect many customers simultaneously, independence is violated and the Poisson model will underestimate variability. A negative binomial model, a non-homogeneous Poisson process, or an incident-based model may then be more suitable.

Outcome and learning points

The analysis gives a transparent baseline for staffing, but it should not be treated as a precise forecast under all conditions. Rate changes by hour, product release, customer type and incident status should be incorporated. The case illustrates three principles:

• distribution choice follows the data-generating process;
• a probability such as P(X ≥ 10) is more useful for planning than a mean alone;
• model diagnostics and operational knowledge are essential.

8. Limitations, Governance and Professional Practice

Probability models can produce misleading decisions when their assumptions are ignored. Common problems include treating dependent observations as independent, using a constant rate during seasonal demand, fitting a normal distribution to severely skewed data, and confusing a modelled probability with certainty.

In business analytics, governance matters. Data used to estimate probabilities should be relevant, sufficiently recent and representative. Customer or employee data may contain privacy-sensitive information and should be processed lawfully, securely and proportionately. Models used in credit, insurance, recruitment or access decisions should be tested for disparate impact and documented so that affected stakeholders can understand the basis of decisions.

Best practice includes:

• define the event and measurement interval precisely;
• state distributional assumptions explicitly;
• visualise data before fitting a model;
• compare predicted and observed frequencies;
• test sensitivity to alternative parameter values;
• monitor model performance after deployment;
• document limitations and retain human oversight for high-impact decisions.

Software packages and AI systems can calculate probabilities quickly, but they do not remove the need for statistical reasoning. An analyst remains responsible for selecting the model, checking inputs, interpreting outputs and communicating uncertainty.

Chapter Summary

Probability distributions model uncertainty in business events. Discrete variables are represented by probability mass functions, while continuous variables are represented by density functions and interval probabilities. The binomial distribution counts successes in a fixed number of independent trials with constant success probability. The Poisson distribution counts events in a fixed interval when events occur independently at a constant average rate. The normal distribution describes continuous, approximately symmetric measurements and uses standardisation through z-scores. The exponential distribution models waiting time between events in a constant-rate Poisson process and has the memoryless property.

The most important professional skill is not merely calculating a probability but selecting and validating a suitable model. Analysts must examine independence, stationarity, skewness, measurement type, data quality and business context. Probabilities support capacity planning, campaign evaluation, service management and risk analysis, but responsible use requires transparency, privacy protection, fairness assessment and ongoing monitoring.

Key Terms

Random variable: A numerical representation of an uncertain outcome.

Discrete random variable: A variable with countable possible values.

Continuous random variable: A variable that can take values across an interval.

Probability mass function: A function giving probabilities for discrete outcomes.

Probability density function: A function whose area over an interval gives a continuous probability.

Cumulative distribution function: The probability that a variable is less than or equal to a specified value.

Expected value: The probability-weighted long-run average of a random variable.

Variance: The average squared distance from the mean.

Standard deviation: The square root of variance.

Binomial distribution: A distribution for successes in a fixed number of independent trials.

Poisson distribution: A distribution for event counts in a fixed interval.

Rate parameter: The expected number of Poisson events per specified interval.

Normal distribution: A symmetric, bell-shaped continuous distribution defined by μ and σ.

Standard normal distribution: A normal distribution with mean 0 and standard deviation 1.

z-score: The number of standard deviations an observation is from the mean.

Exponential distribution: A distribution for waiting time between events in a constant-rate process.

Memoryless property: The exponential property that elapsed waiting does not change future waiting-time probabilities.

Independence: A condition in which the occurrence of one event does not alter another event’s probability.

Model assumption: A condition required for a probability model to be appropriate.

Review Questions

Short Answer Questions

1. Distinguish between a discrete and a continuous random variable, giving one business example of each.
2. State the five main conditions required for a binomial model.
3. What does the Poisson parameter λ represent?
4. Explain the meaning of a z-score of −2.
5. Distinguish between a Poisson count and an exponential waiting time.

Long Answer Questions

1. Explain the binomial distribution, derive its mean and variance conceptually, and discuss two business applications.
2. Describe the assumptions of the Poisson distribution and explain how violations could affect service-capacity decisions.
3. Explain how standardisation is used with the normal distribution. Include a worked business example.
4. Compare the normal and exponential distributions in terms of shape, possible values, parameters and applications.
5. Discuss why model validation, governance and ethical review are important when probability distributions are used in business decisions.

Analytical / Application Questions

1. A campaign contacts 30 customers, each having an independently estimated purchase probability of 0.08. Identify the appropriate distribution and calculate the expected number of purchases and the probability of no purchases.
2. A help desk receives 3 requests per hour on average. Use a suitable model to calculate the probability of at least one request in 30 minutes and explain the operational meaning.
3. Product fill volume is approximately normal with mean 500 millilitres and standard deviation 12 millilitres. Calculate the proportion expected to lie between 476 and 524 millilitres, and evaluate whether a normal model is plausible if fill volumes cannot be negative but are otherwise approximately symmetric.

Multiple Choice Questions with Answers

Question 1

Which situation is most appropriately modelled by a binomial distribution?

A. The number of customers arriving in one hour

B. The number of purchases among 20 independently contacted customers

C. The time until the next customer arrives

D. The weight of a randomly selected parcel

Correct Answer: B. The number of purchases among 20 independently contacted customers

Question 2

If X follows a binomial distribution with n = 50 and p = 0.20, what is E(X)?

A. 2

B. 10

C. 25

D. 250

Correct Answer: B. 10

Question 3

A Poisson random variable has parameter λ = 4. What are its mean and variance?

A. Mean 2 and variance 4

B. Mean 4 and variance 2

C. Mean 4 and variance 4

D. Mean 16 and variance 4

Correct Answer: C. Mean 4 and variance 4

Question 4

A process averages 12 events per hour. Under a Poisson model, what is the expected number of events in 15 minutes?

A. 0.25

B. 3

C. 12

D. 48

Correct Answer: B. 3

Question 5

If a variable has mean 100 and standard deviation 15, what is the z-score for an observation of 130?

A. −2

B. 0.5

C. 2

D. 15

Correct Answer: C. 2

Question 6

Approximately what percentage of observations lie within two standard deviations of the mean in a normal distribution?

A. 50%

B. 68%

C. 95%

D. 99.7%

Correct Answer: C. 95%

Question 7

Which distribution is most suitable for the waiting time until the next event in a constant-rate Poisson process?

A. Binomial

B. Poisson

C. Normal

D. Exponential

Correct Answer: D. Exponential

Question 8

Customers arrive at a rate of 6 per hour. What is the mean waiting time between arrivals?

A. 6 hours

B. 1 hour

C. 10 minutes

D. 36 minutes

Correct Answer: C. 10 minutes

Question 9

Which assumption is especially important for the standard Poisson model?

A. The variable must be negative

B. Events occur at a constant average rate and independently

C. There must be exactly two outcomes per trial

D. The distribution must be symmetric

Correct Answer: B. Events occur at a constant average rate and independently

Question 10

For a continuous random variable, what is the probability of an exact single value under a continuous distribution?

A. Always 1

B. Always 0

C. Equal to the mean

D. Equal to the standard deviation

Correct Answer: B. Always 0