Probability Distribution

A probability distribution shows the full picture of uncertainty. For a simple example, the distribution for rolling a fair die gives each number from 1 to 6 a probability of one-sixth.

Some distributions are especially common. The normal distribution, shaped like a bell, describes many natural phenomena and is often used as an approximation for investment returns. Most outcomes cluster around the average, with fewer outcomes further away. The binomial distribution describes the number of successes in a fixed number of trials, such as how many of 100 loans default. The lognormal distribution is often used for asset prices, which cannot fall below zero.

Distributions are summarised by measures such as the mean, which shows the centre; the standard deviation, which shows the spread; skewness, which shows whether the distribution leans to one side; and kurtosis, which shows how heavy its tails are.

In finance, understanding the distribution of returns is crucial for managing risk. A common problem is assuming returns follow a normal distribution when, in reality, extreme outcomes occur more often. Models built on this assumption may underestimate the chance of large losses.