Level 1 CFA® Exam:
Probability Distributions – Introduction
When facing investment decisions, we often encounter random variables. One such variable is return on investment. To determine how likely a given value of a random variable is, we need to know the probability distribution of the random variable. Just to remind you and make things fall into place, let's revise some basic concepts.
A random variable is a variable that can take different numerical values depending on the situation. One example of a random variable will be the outcome of a dice roll. When rolling a dice we don't know what number we're going to get. What's more, the number is random and it depends on chance.
The set of all possible values or outcomes that a random variable can take is called a sample space. In the case of a dice roll, the possible outcomes include the numbers 1, 2, 3, 4, 5, and 6.
To each value that a random variable within a sample space can take, we can assign its probability. This is a function we call probability distribution.
We distinguish between:
- discrete random variables and
- continuous random variables.
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The probability function helps us determine the probability of a random variable taking on a given value. The notation \(P(X=x)\) means that a random variable 'X' takes on the value of 'x'.
For a discrete random variable, the probability function (aka. probability mass function) is labelled \(p(x)\).
In the case of a continuous random variable, the probability is labeled \(f(x)\) and the probability function is called a probability density function or simply density.
Key Characteristics of Probability Function
There are two key characteristics of the probability function:
- The probability of any event is always equal to or greater than 0 and less than or equal to 1. If an event is impossible, its probability equals 0 and if an event is certain, its probability is 1. If an event may or may not occur, its probability is somewhere between 0 and 1.
- The sum of the probabilities of all possible values that the random variable X can take on equals 1.
Back to the example with a dice roll. You know now that we're dealing here with a discrete variable. Let's write down the probability for every possible value of the variable X:
| \(x\) | \(p(x)\) |
|---|---|
| \(1\) | \(\frac{1}{6}\) |
| \(2\) | \(\frac{1}{6}\) |
| \(3\) | \(\frac{1}{6}\) |
| \(4\) | \(\frac{1}{6}\) |
| \(5\) | \(\frac{1}{6}\) |
| \(6\) | \(\frac{1}{6}\) |
Each possible number of dots has the same probability of appearing on the top face of a dice. In the example, the numbers 1, 2, 3, 4, 5, and 6 make up the set of possible values that the variable can take on.
The probability function of a random variable only shows us the probability that a variable takes on a given value. Sometimes, however, we want to know the probability that a random variable is greater or lower than a certain value x. This is when we can use cumulative probabilities.
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Level 1 CFA Exam Takeaways: Probability Distribution – Introduction
star content check off when done- A random variable is a variable that can take different numerical values depending on the situation.
- The set of all possible values or outcomes that a random variable can take is called a sample space.
- We distinguish between discrete random variables and continuous random variables.
- A discrete random variable (aka. discrete variable), is a random variable that can only take on values from a countable set of possible outcomes.
- A continuous random variable can take on infinitely many values.
- The probability function helps us determine the probability of a random variable taking on a given value.
- For a discrete random variable, the probability function (aka. probability mass function) is labelled \(p(x)\).
- For a continuous random variable, the probability function (aka. probability density function) is labelled \(f(x)\).
- The cumulative distribution function tells us what the probability that the random variable X takes on the value less than or equal to some value x is.
- The cumulative distribution function is denoted as \(F(x)\).