1 Basic Notation
1.1 Probabilities
A quick note on notation before we begin. When we talk about probabilities, we will denote the probability of a particular outcome as:
\[ P(outcome) \]
A value of 0.5 indicates a probability of 50%. If we were to repeat the experiment many many times, we would expect this outcome to occur in about half of our trials.
Sometimes we will want to get more specific and indicate what we are measuring and what outcomes are possible. We could use \(X\) to indicate a random variable with different possible outcomes. Let’s call these \(outcome_1\), \(outcome_2\), and so on. The probability of \(X\) taking on a specific outcome can also be written as:
\[ P(X=outcome_1) \]
If we know all possible outcomes, the probabilities should sum to 1. We can express this using the sum below which iterates over each outcome \(i\) out of \(I\) possible outcomes:
\[ \sum\limits_{i=1}^I P(X=outcome_i) = 1 \]
We may also want to specify a probability conditional upon some kind of assumption or model. This would be indicated as:
\[ P(X=outcome_i|A) \]
or just
\[ P(outcome_i|A) \]
Both of these mean the same thing. They express the probability that the random variable \(X\) (which is implied in the second form but not explicitly called out) will be \(outcome_i\) given a set of assumptions \(A\).
Other notation and concepts will be introduced as we need it. This should be enough to jump into the first chapter without confusion.
1.2 Expected Values
The “expected value” is the mean or average outcome that we expect in a numerical variable. This is indicated as:
\[ E(X) \]
As an example, if we number the sides of a die from 1 to 6 and roll the die repeatedly, the expected value is 3.5 (the average of all numbers from 1 to 6). This isn’t actually one of the specific outcomes. It is simply an average that gives you some idea of the location or middle outcome or the probability distribution.
We can also indicate the variance of this numerical outcome as:
\[ var(X) \]
This has the same meaning as the variance in statistics.
We may also use notation to indicate the mean and standard deviation of a random variable \(X\) with:
- \(\mu_X\) = mean of X
- \(\sigma_X\) = standard deviatino of X
- \(\sigma_X^2\) = variance of X