Statistics for Analytics

Chapter 7
Random Variables and Probability Distributions

Diploma in Business Analytics
Statistics for Analytics   •   Diploma

Random Variables and Probability Distributions

Introduction

Business analytics frequently deals with uncertain outcomes: whether a customer purchases, how many claims arrive in an hour, the time until a service request is answered, or the amount spent in a transaction. Probability distributions provide a systematic way to represent this uncertainty. They allow analysts to calculate the likelihood of events, estimate expected outcomes, assess risk and select appropriate analytical models.

A random variable is a numerical representation of the outcome of a random process. For example, the number of products purchased by a customer can be represented by X, while the waiting time for a call-centre response can be represented by T. A probability distribution describes the possible values of a random variable and the probabilities associated with them.

This chapter develops the major distributions used in introductory business analytics: Bernoulli, binomial, Poisson, uniform, normal and exponential distributions. It also explains their assumptions, parameters, calculations, applications and limitations.

Learning Objectives

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

  1. Distinguish between discrete and continuous random variables.
  2. Interpret probability mass functions, probability density functions and cumulative distribution functions.
  3. Calculate expected values and variances for common distributions.
  4. Apply Bernoulli, binomial and Poisson models to discrete business outcomes.
  5. Apply uniform, normal and exponential models to continuous outcomes.
  6. Select a suitable probability distribution for a business problem.
  7. Interpret distribution results responsibly in operational and managerial decisions.

1. Random Variables and Their Distributions

1.1 Discrete and continuous random variables

A discrete random variable takes countable values, usually integers. Examples include:

  • number of defective items in a batch;
  • number of website orders in an hour;
  • number of customers who respond to an email campaign.

A continuous random variable can take any value within an interval. Examples include:

  • delivery time;
  • transaction value;
  • temperature in a warehouse;
  • duration of a customer-service call.

The distinction matters because discrete variables are described using individual probabilities, whereas continuous variables are described using areas under a curve. For a continuous variable, the probability of one exact value is normally zero; probabilities concern intervals such as P(10 ≤ T ≤ 15).

1.2 Probability mass, density and distribution functions

For a discrete random variable X, the probability mass function (PMF) is

p(x) = P(X = x)

The PMF must satisfy

p(x) ≥ 0, ∑ p(x) = 1

where the summation is over all possible values of X.

For a continuous random variable, the probability density function (PDF) is written f(x). It must be non-negative and have total area one:

f(x) ≥ 0, ∫−∞∞ f(x) dx = 1

Probabilities are areas under the density curve:

P(a ≤ X ≤ b) = ∫ab f(x) dx

The cumulative distribution function (CDF) applies to both discrete and continuous variables:

F(x) = P(X ≤ x)

It is non-decreasing and ranges from 0 to 1. In practice, software commonly provides CDF values, quantiles and random samples for standard distributions.

1.3 Expected value and variance

The expected value is the long-run average value of a random variable. For a discrete variable,

E(X) = μ = ∑ x p(x)

For a continuous variable,

E(X) = μ = ∫−∞∞ x f(x) dx

The expected value need not be an outcome that can actually occur. For example, an expected number of daily complaints may be 12.4 even though a particular day must have an integer number of complaints.

Variance measures dispersion around the mean:

Var(X) = σ² = E[(X − μ)²]

For a discrete variable, an equivalent computational form is

Var(X) = E(X²) − [E(X)]²

The standard deviation is σ = √(σ²), expressed in the same units as X. Variance is measured in squared units, so standard deviation is generally easier to interpret.

For a linear transformation Y = a + bX,

E(Y) = a + bE(X), Var(Y) = b² Var(X)

The variance formula shows why the multiplier is squared. These relationships assume that a and b are constants.

2. The Bernoulli Distribution

A Bernoulli trial has exactly two possible outcomes, conventionally called success and failure. Let X = 1 represent success and X = 0 represent failure. If the probability of success is p, then the probability of failure is 1 − p.

The Bernoulli PMF is

P(X = x) = px(1 − p)1−x, x ∈ {0,1}

The mean and variance are

E(X) = p
Var(X) = p(1 − p)

The model applies when there is one trial, two mutually exclusive outcomes, and a fixed probability of success. A customer either renews or does not renew; a payment either succeeds or fails. The label “success” does not imply a desirable outcome—it simply identifies the outcome coded as 1.

Example. If a customer has probability 0.30 of purchasing an extended warranty, then the expected value of the purchase indicator is 0.30. Across many similar customers, approximately 30% are expected to purchase. The variance is 0.30 × 0.70 = 0.21.

3. The Binomial Distribution

The binomial distribution counts successes in a fixed number of Bernoulli trials. Let X be the number of successes in n trials, with success probability p on each trial. The assumptions are:

  1. The number of trials n is fixed.
  2. Each trial has two outcomes.
  3. Trials are independent, or are treated as approximately independent.
  4. The success probability p is constant across trials.

The probability of exactly x successes is

P(X = x) = C(n,x) px(1 − p)n−x, x = 0,1,…,n

where

C(n,x) = n! / [x!(n−x)!]

and n! is the factorial of n. The mean and variance are

E(X) = np
Var(X) = np(1 − p)

Worked example. A retailer estimates that 20% of five targeted customers will respond to an offer. Assuming independent responses, the probability of exactly two responses is

P(X = 2) = C(5,2)(0.20)²(0.80)³

Since C(5,2) = 10,

P(X = 2) = 10 × 0.04 × 0.512 = 0.2048

Thus, the probability is 0.2048, or 20.48%. The expected number of responses is 5 × 0.20 = 1, and the variance is 5 × 0.20 × 0.80 = 0.80.

The binomial model is unsuitable if customers influence one another, probabilities vary substantially across customers, or the number of opportunities is not fixed. For sampling without replacement from a small population, a hypergeometric distribution may be more appropriate.

4. The Poisson Distribution

The Poisson distribution models the number of events occurring in a fixed interval of time, area, length or volume. Let X be the event count and λ be the average number of events per interval. Its PMF is

P(X = x) = e−λ λx / x!, x = 0,1,2,…

Here, e ≈ 2.71828. For a Poisson random variable,

E(X) = λ
Var(X) = λ

The assumptions are that events occur singly, the average rate is stable over the interval, occurrences in non-overlapping intervals are independent, and the probability of two events occurring at exactly the same instant is negligible.

If the rate is r events per unit and the interval has length t, then λ = rt. Rates must be converted to compatible units.

Worked example. A help desk receives an average of three urgent tickets per hour. The probability of exactly two urgent tickets in one hour is

P(X = 2) = e−3 3² / 2! ≈ 0.2240

The expected number and standard deviation are respectively 3 tickets and √3 ≈ 1.732 tickets per hour.

For independent Poisson counts, the total count is also Poisson: if X₁ and X₂ have means λ₁ and λ₂, then X₁ + X₂ has mean λ₁ + λ₂. This is useful when combining incoming events from separate sources, provided independence is credible.

The Poisson model can approximate a binomial distribution when n is large, p is small and λ = np. It should not be used automatically when data show clustering, changing rates or overdispersion, where negative binomial or time-varying models may be better.

5. Continuous Distributions

5.1 Uniform distribution

A continuous variable has a uniform distribution on [a,b] if all values in the interval are equally dense. Its PDF is

f(x) = 1/(b−a), a ≤ x ≤ b

and zero elsewhere. The mean and variance are

E(X) = (a+b)/2
Var(X) = (b−a)²/12

For c,d within [a,b],

P(c ≤ X ≤ d) = (d−c)/(b−a)

Example. If a delivery is equally likely to arrive between 20 and 30 minutes, the probability of arrival between 24 and 27 minutes is (27−24)/(30−20) = 0.30. The expected delivery time is 25 minutes.

Uniformity is often a simplifying assumption used when only minimum and maximum values are known. It is inappropriate when values cluster around a central point or when the process has a clear trend.

5.2 Normal distribution

The normal distribution is continuous, symmetric and bell-shaped. It is determined by mean μ and standard deviation σ > 0. Its PDF is

f(x) = [1/(σ√(2π))] exp{−(x−μ)²/(2σ²)}, −∞ < x < ∞

The normal distribution is widely used because measurement errors and aggregate business outcomes often approximate it. However, not every business variable is normal: sales amounts may be skewed, counts are discrete, and times cannot be negative.

A value can be standardized using the z-score:

Z = (X−μ)/σ

If X is normally distributed, then Z follows the standard normal distribution with mean 0 and standard deviation 1. Therefore,

P(X ≤ x) = P(Z ≤ (x−μ)/σ)

Worked example. Suppose daily demand is approximately normal with mean 500 units and standard deviation 80 units. The probability that demand is at most 620 units is found by

z = (620−500)/80 = 1.5

The standard normal CDF gives P(Z ≤ 1.5) ≈ 0.9332. Thus, approximately 93.32% of days have demand no greater than 620 units. This result is useful when assessing a proposed inventory capacity, although the analyst should also consider forecast error, seasonality and the impossibility of negative demand.

The empirical rule states that for a normal distribution approximately 68% of observations lie within one standard deviation of the mean, 95% within two, and 99.7% within three. These are approximations, not universal laws.

For a continuous normal variable, the probability of exactly the mean is zero. In practical applications, measured values are often rounded, so interval probabilities and continuity corrections may be relevant.

5.3 Exponential distribution

The exponential distribution models a non-negative waiting time until the next event in a Poisson process. If the event rate is λ > 0 per unit time, its PDF is

f(t) = λe−λt, t ≥ 0

The CDF and survival probability are

P(T ≤ t) = 1 − e−λt
P(T > t) = e−λt

The mean and variance are

E(T) = 1/λ
Var(T) = 1/λ²

The exponential distribution has the memoryless property:

P(T > s+t | T > s) = P(T > t)

This means that the remaining waiting time does not depend on how long the process has already waited. This assumption is often unrealistic for equipment that deteriorates with age or customers whose patience changes over time.

Example. If service requests arrive at a rate of six per hour, the mean time between requests is 1/6 hour, or 10 minutes. The probability that the next request arrives within five minutes is obtained by using t = 5/60 hour:

P(T ≤ 5/60) = 1 − e−6(5/60) = 1 − e−0.5 ≈ 0.3935

Therefore, there is approximately a 39.35% probability that the next request arrives within five minutes.

6. Selecting a Suitable Distribution

Distribution selection should begin with the business process, not with a preferred formula. Ask the following questions:

QuestionLikely model
Is there one trial with two outcomes?Bernoulli
Is the number of successes counted in a fixed number of trials?Binomial
Is the number of arrivals counted in a fixed interval?Poisson
Is every value between two limits equally plausible?Uniform
Is the outcome continuous and approximately symmetric and bell-shaped?Normal
Is the waiting time until the next event being modelled?Exponential

Analysts should inspect data using histograms, frequency tables, box plots and summary statistics. Important diagnostic questions include:

  • Is the outcome discrete or continuous?
  • Are observations independent?
  • Is the rate stable?
  • Is the variable bounded or unbounded?
  • Is the distribution skewed?
  • Are there outliers, seasonal effects or subgroups?
  • Does the model produce plausible values?

A model is an approximation, not a statement that nature must follow an ideal mathematical curve. Good practice includes comparing predicted and observed frequencies, testing sensitivity to parameter changes and documenting assumptions.

7. Applied Case: Capacity Planning for a Customer-Service Centre

Background

This illustrative case concerns a digital bank planning staffing for its urgent-support queue. Historical data suggest an average of six urgent requests per hour during a particular operating period. Management wants to estimate arrival risk and choose a staffing capacity.

Problem

The operations team asks:

  1. What is the probability of receiving exactly four requests in an hour?
  2. What is the probability of receiving more than eight requests?
  3. What is the probability that the next request arrives within ten minutes?

Analysis and approach

Because the questions concern event counts in a fixed time interval and waiting time to the next event, the Poisson and exponential distributions are natural starting points. Let X be the number of requests per hour, with λ = 6. Let T be the time to the next request, measured in hours.

For exactly four requests,

P(X = 4) = e−66⁴/4! ≈ 0.1339

For more than eight requests,

P(X > 8) = 1 − P(X ≤ 8)

Using the Poisson CDF, P(X ≤ 8) ≈ 0.8472, so

P(X > 8) ≈ 1 − 0.8472 = 0.1528

There is therefore approximately a 15.28% chance of more than eight requests in an hour.

For the next request within ten minutes, convert ten minutes to one-sixth of an hour:

P(T ≤ 1/6) = 1 − e−6(1/6) = 1 − e−1 ≈ 0.6321

Outcome and decision implications

The results indicate that high arrival volumes are not rare enough to ignore. A manager might schedule flexible capacity, create escalation procedures or use a service-level simulation. However, the calculations do not directly determine the number of agents required. That decision also depends on service time, agent productivity, queue discipline, abandonment and the required response standard. A queueing model would be needed for a complete staffing analysis.

Learning points

The case demonstrates that distribution choice follows the measurement question: Poisson models counts, whereas exponential models waiting times. It also shows why a mathematically correct probability can still be insufficient for a managerial decision if important operational variables are omitted.

8. Technology, Governance and Limitations

Spreadsheet functions and statistical software can calculate PMFs, CDFs, quantiles and simulated observations. For example, software may provide a Poisson CDF, a normal quantile or random samples for scenario analysis. Automation reduces arithmetic errors but does not validate assumptions. Analysts remain responsible for selecting the distribution, checking units and interpreting results.

AI systems can assist with exploratory analysis, code generation and model comparison. They can also confidently recommend an unsuitable distribution. Any AI-supported analysis should therefore include data checks, reproducible code, parameter documentation, independent validation and human review.

Probability models may affect staffing, credit, insurance, fraud detection and customer treatment. Governance requires transparent assumptions, appropriate access controls, privacy protection and monitoring for biased outcomes. A distribution estimated from historical data may reproduce historical inequities or fail when economic conditions change. Probabilities should not be presented as certainties, and decision-makers should understand uncertainty in the parameters as well as randomness in individual outcomes.

Common limitations include:

  • Independence failure: repeated observations may be correlated.
  • Non-constant rates: demand may vary by hour, day or season.
  • Heterogeneity: one population may contain several different customer groups.
  • Boundaries: normal models can assign probability to impossible negative values.
  • Parameter uncertainty: estimated means and rates are not known exactly.
  • Model risk: a convenient distribution may fit historical data but fail under new conditions.

Best practice is to combine theoretical reasoning with empirical diagnostics, sensitivity analysis and periodic model review.

Chapter Summary

A random variable assigns numerical values to uncertain outcomes. Discrete variables are described by probability mass functions, while continuous variables are described by density functions and interval areas. The cumulative distribution function gives the probability of observing a value no greater than a specified threshold.

The expected value describes long-run average behaviour, and variance and standard deviation describe dispersion. The Bernoulli distribution models one binary trial; the binomial distribution counts successes in a fixed number of independent trials; and the Poisson distribution counts events in a fixed interval under a stable-rate assumption. The uniform distribution represents equal density over a bounded interval. The normal distribution models continuous, approximately symmetric outcomes, and the exponential distribution models waiting time to the next event in a Poisson process.

Distribution selection must reflect the structure of the business process. Analysts should check independence, measurement scale, rate stability, bounds, skewness and data fit. Software and AI tools support calculation and simulation but do not replace judgement, validation or governance. Probability results become professionally useful when their assumptions and managerial implications are communicated clearly.

Key Terms

Random variable: A numerical quantity whose value depends on the outcome of a random process.

Discrete random variable: A random variable with countable possible values.

Continuous random variable: A random variable that can take any value in an interval.

Probability mass function: A function giving probabilities for the values of a discrete variable.

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

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

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

Variance: The expected squared deviation from the mean.

Standard deviation: The square root of variance.

Bernoulli distribution: A distribution for one trial with two possible outcomes.

Binomial distribution: A distribution for the number of successes in fixed Bernoulli trials.

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

Uniform distribution: A distribution with constant density over a specified interval.

Normal distribution: A symmetric bell-shaped continuous distribution defined by a mean and standard deviation.

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

Z-score: The number of standard deviations an observation lies from the mean.

Exponential distribution: A distribution for waiting time until the next event in a Poisson process.

Rate parameter: The average number of events per unit interval, commonly denoted λ.

Memoryless property: The property that future exponential waiting time does not depend on elapsed waiting time.

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

Review Questions

Short Answer Questions

  1. What is the difference between a discrete and a continuous random variable?
  2. State the mean and variance of a Bernoulli random variable with success probability p.
  3. What does the parameter λ represent in a Poisson distribution?
  4. Why is the probability of one exact value normally zero for a continuous random variable?
  5. What does a z-score measure?

Long Answer Questions

  1. Explain the assumptions of the binomial distribution and discuss two business situations in which they may fail.
  2. Compare the Poisson and exponential distributions, including the types of questions each answers.
  3. Explain how expected value, variance and standard deviation support business decision-making.
  4. Describe the procedure for using a normal distribution to calculate a probability for a raw observation.
  5. Discuss why distribution selection should combine theoretical knowledge with empirical data analysis.

Analytical / Application Questions

  1. A retailer estimates that 15% of customers redeem a digital coupon. For 10 independent customers, calculate the probability that exactly two redeem it and the expected number of redemptions.
  2. A warehouse receives an average of four urgent orders per hour. Using a Poisson model, calculate the probability of no urgent orders in 30 minutes and explain one operational implication.
  3. A delivery time is modelled as normal with mean 40 minutes and standard deviation 6 minutes. Calculate the probability that a delivery takes more than 50 minutes and explain one limitation of applying a normal model to delivery times.

Multiple Choice Questions with Answers

Question 1

Which variable is discrete?

A. Customer waiting time in minutes

B. Number of complaints received in a day

C. Temperature of a warehouse

D. Transaction value in dollars

Correct Answer: B. Number of complaints received in a day

Question 2

A Bernoulli trial has success probability 0.7. What is its variance?

A. 0.07

B. 0.21

C. 0.49

D. 0.70

Correct Answer: B. 0.21

Question 3

Which situation is most appropriately modelled by a binomial distribution?

A. The time until the next website visit

B. The number of defective items in a fixed sample of independently inspected items

C. The temperature recorded continuously during a day

D. The average delivery time

Correct Answer: B. The number of defective items in a fixed sample of independently inspected items

Question 4

A Poisson random variable has mean 8. What is its variance?

A. 2

B. 4

C. 8

D. 64

Correct Answer: C. 8

Question 5

If events occur at a rate of 6 per hour, what is the mean waiting time until the next event?

A. 6 hours

B. 1 hour

C. 1/6 hour

D. 36 hours

Correct Answer: C. 1/6 hour

Question 6

A continuous variable is uniformly distributed between 10 and 20. What is the probability that it lies between 12 and 15?

A. 0.15

B. 0.30

C. 0.50

D. 0.70

Correct Answer: B. 0.30

Question 7

A normal variable has mean 100 and standard deviation 15. What is the z-score for 130?

A. 1

B. 1.5

C. 2

D. 3

Correct Answer: C. 2

Question 8

Which assumption is associated with the exponential distribution in its standard waiting-time application?

A. The waiting time is always symmetric around its mean

B. The process has a constant event rate

C. The variable can take negative values

D. There are exactly two outcomes

Correct Answer: B. The process has a constant event rate

Question 9

If X has expected value 10, what is the expected value of Y = 5 + 3X?

A. 15

B. 30

C. 35

D. 50

Correct Answer: C. 35

Question 10

Which question is best answered by a Poisson distribution?

A. What is the probability that exactly five calls arrive in an hour?

B. What is the probability that a customer's income exceeds $50,000?

C. What is the average weight of a parcel?

D. What is the probability that one particular customer renews?

Correct Answer: A. What is the probability that exactly five calls arrive in an hour?
Statistics for Analytics
Chapter 7 of 14