**Probability Distribution**

The probability distribution of a random variable “X” is basically a **graphical presentation** of the probabilities of all possible outcomes of X. A random variable is any quantity for which more than one value is possible, for instance, the price of quoted stocks. Simply put, a probability distribution gathers all the outcomes and goes a step further to indicate the probability associated with each outcome.

Example: Probability distribution

Suppose we roll a dice. The set of possible outcomes is:

{1 2 3 4 5 6}

Each of these outcomes would occur with a probability of 1/6 because each outcome has an equal chance of occurring. Consequently, the probability distribution would be a straight line:

*Note to candidates: Although the above distribution is a straight line, most real life distributions are usually curved. The CFA curriculum particularly delves into the bell-shaped normal distribution.*

**Discrete random variables**

A discrete random variable is one that can take on a finite number of outcomes.

Examples:

- If we roll a dice, there are 6 possible outcomes. Therefore, the outcomes are discrete and random.
- The number of CFA finalists employed within a given year is a discrete random variable

**Continuous Random variable**

A continuous random variable is that which has an infinite number of possible outcomes. A good example can be the rate of return on a stock. For instance, the return can be 6% or between 6% and 7%, in which case it can take on 6.4%, 6.41%, 6.412% or even 6.412325% i.e. infinite values.

**Probability Function**

A probability function gives the probability of a random variable X taking on a value “x”. The probability functions of discrete and continuous random variables are slightly different:

For a discrete random variable, the probability function, P(x), satisfies the following properties:

- P(X = x) = P(x)

In statistics, P(x) is said to be a probability mass function - P(x) is always nonnegative for all x
- The sum of the probabilities of all possible outcomes = 1.

For a continuous random variable, the probability function, f(x), satisfies the following properties:

- The probability that x is between two values, a and b is

$$ P\left\{ a\le x\le b \right\} =\int _{ a }^{ b }{ f\left( x \right) dx } $$

f(x) is said to be a probability density function(pdf) - f(x) is nonnegative for all x
- The integral of the probability function is 1, that is

$$ \int _{ -\infty }^{ \infty }{ f\left( x \right) dx=1 } $$

*Reading 9 LOS 9a:*

*Define a probability distribution and distinguish between discrete and continuous random variables and their probability functions.*