28 Probability Distributions
A probability distribution describes how the probabilities of a random variable are spread across its possible values. Distributions come in two broad families: discrete distributions, for variables that take countable values (such as the number of defective items), and continuous distributions, for variables that can take any value in a range (such as height or time). This chapter focuses on the two most widely used distributions in business analytics: the Binomial (discrete) and the Normal (continuous).
28.1 Discrete vs Continuous Random Variables
- Discrete Random Variable: Takes a countable number of distinct values (0, 1, 2, …), each with its own probability. Example: the number of customers who default on a loan out of 20.
- Continuous Random Variable: Can take any value within an interval, with probability described by a density curve rather than a single point probability. Example: the exact weight of a manufactured item.
28.2 The Binomial Distribution
The binomial distribution models the number of “successes” in a fixed number of independent trials, each with the same probability of success — for example, the number of defective items in a batch, or the number of customers who click on an ad out of those shown it.
Conditions for a binomial situation:
- A fixed number of trials, \(n\).
- Each trial has only two possible outcomes (success/failure).
- The probability of success, \(p\), is the same on every trial.
- The trials are independent of each other.
The probability of exactly \(x\) successes is: \[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}, \qquad x = 0, 1, 2, \ldots, n \]
Mean and variance of the binomial distribution: \[ \mu = np \qquad \sigma^2 = np(1-p) \]
28.2.1 Binomial Worked Example
A call centre knows that 20% of calls result in a successful sale. Out of the next 10 calls, what is the probability that exactly 3 result in a sale?
Here \(n = 10\), \(p = 0.20\), \(x = 3\).
\[ P(X = 3) = \binom{10}{3} (0.20)^3 (0.80)^7 = 120 \times 0.008 \times 0.2097 \approx 0.2013 \]
There is approximately a 20.1% chance of exactly 3 sales out of the next 10 calls.
28.3 The Normal Distribution
The normal distribution is the most important continuous distribution in statistics: a symmetric, bell-shaped curve fully described by its mean \(\mu\) and standard deviation \(\sigma\). Many natural and business measurements — heights, test scores, measurement errors — approximate a normal distribution, and it underlies most confidence intervals and hypothesis tests used later in this book.
Its probability density function is: \[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} \, e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]
A normal distribution with \(\mu = 0\) and \(\sigma = 1\) is called the standard normal distribution, and any normal variable can be converted to it using the z-score: \[ z = \frac{x - \mu}{\sigma} \]
The empirical (68-95-99.7) rule: approximately 68% of values fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
28.3.1 Normal Distribution Worked Example
Exam scores are normally distributed with a mean of 70 and a standard deviation of 10. What proportion of students score above 85?
\[ z = \frac{85 - 70}{10} = 1.5 \]
From the standard normal table, \(P(Z \le 1.5) \approx 0.9332\), so \(P(Z > 1.5) = 1 - 0.9332 = 0.0668\).
Approximately 6.7% of students score above 85.
28.4 Binomial and Normal Distributions in R and Python
Transition to Sampling Distributions
The binomial and normal distributions describe how individual observations behave. But inferential statistics is built on the behaviour of sample statistics — such as a sample mean — computed over and over from repeated samples. The next chapter introduces sampling distributions and the Central Limit Theorem, which explain why the normal distribution shows up so often in inference even when the underlying data is not normally distributed.
Summary
| Concept | Description |
|---|---|
| Foundations | |
| Discrete Random Variable | A variable that takes a countable number of distinct values, each with its own probability |
| Continuous Random Variable | A variable that can take any value within an interval, described by a density curve |
| Binomial Distribution | |
| Binomial Distribution | Models the number of successes in a fixed number of independent trials with constant success probability |
| Binomial Conditions | Fixed number of trials, two outcomes per trial, constant success probability, independent trials |
| Binomial PMF | P(X=x) equals n-choose-x times p to the x times (1-p) to the (n-x) |
| Binomial Mean and Variance | Mean equals n times p; variance equals n times p times (1-p) |
| Normal Distribution | |
| Normal Distribution | A symmetric, bell-shaped continuous distribution defined by its mean and standard deviation |
| Normal PDF | The density function of the normal distribution, involving the mean, standard deviation, and e |
| Standard Normal Distribution | A normal distribution with mean 0 and standard deviation 1 |
| Z-score | Standardised value showing how many standard deviations an observation is from the mean |
| Empirical (68-95-99.7) Rule | Approximately 68%, 95%, and 99.7% of values fall within 1, 2, and 3 standard deviations of the mean |
| In R and Python | |
| R dbinom()/pnorm() | R functions used to compute binomial probabilities and normal cumulative probabilities |
| Python scipy.stats binom/norm | SciPy distribution objects used to compute binomial and normal probabilities in Python |