Introduction to Continuous Probability Distributions (Business)

The Probability Density Function

Since a continuous random variable can take any value within its range, we cannot list all the possible values and their probabilities as in the discrete case (see pages on discrete distributions). For instance the rate of inflation could be recorded to any number of decimal places so it is impossible to list all possible inflation rates.

For continuous random variables we represent probabilities using a probability density function (pdf) (sometimes just called the probability distribution). The pdf of a continuous random variable $X$, is a function $f(x)$ defined such that:

• Its curve lies on or above the $x$-axis, i.e. $f(x) \ge 0$ for all $x$ in its range.
• The area under the entire curve is $1$.
• The probability $\mathrm{P}(a<X<b)$ that $X$ lies between $a$ and $b$ is the area under the curve between $a$ and $b$.

Below is an example of what a pdf might look like. The region shaded green is $\mathrm{P}(2 \leq X \leq 3)$. The total shaded area (purple and green) is equal to $1$.

Cumulative Probabilities

As for discrete random variables, if $X$ is a random variable then the cumulative probability ($P(X \leq 10)$ for example) is the cumulative probability that $X$ takes any value less than or equal to $10$.

The cumulative distribution function (cdf) gives the probability that the random variable $X$ is less than or equal to $x$ (i.e the cumulative probability) and is usually denoted $F(x)$. It is equal to the area under the curve of the pdf $f(x)$.

The Expected Value and Variance

In general, calculating the expected value (mean) and variance of a continuous random variable requires using integration (not covered here). However, the continuous probability distributions you will be dealing with in this section have well known formulae for the means and variances and we use these formulae.

Test Yourself

Click on the following links to practise Numbas tests on the distributions on this page:

Test yourself: Numbas test on calculating probabilities from a normal distribution

Test yourself: Numbas test on the exponential distribution and uniform distribution