2.6.4 (Page 123) ~ 2.34, 2.36, 2.38, 2.39
The mean or expected value of a random variable can be thought of as the “long-term average”, meaning the average of the outcomes of an ever increasing number of trials of the experiment.
Denoted $E(X)$ or $\mu_X$.
If the variable $X$ takes the values $x_1, x_2, \ldots x_n$ with probabilities $p_1, p_2, \ldots, p_n$ respectively, then the mean is defined as:
You can think of this as a weighted average of the values, with their probabilities as weights. This makes sense: We want to take all values into account, but those values that have a higher probability are meant to appear more often, and so should contribute more. Each value contributes an amount proportionate to its relative frequency.
As a simple example, consider the example from the last section, with probability table:
X 0 1 2 P(X) $1/2$ $1/4$ $1/4$ ----- ------ ------ ------
Then for the mean we would have:
We can think of this as saying that if you were to play that game repeatedly, you would be gaining on average $$0.75$ per game. You can also think of it as the “fair price to pay to play the game”.
We examined a number of games in the previous section. Compute the mean of the random variables in each of those games.
The standard deviation follows a similar formula:
So we look at how far each value is from the mean, square to remove the signs, average while accounting for the different probabilities, and finally take a square root.
This square of the standard deviation, typically called the Variance $\textrm{Var}(X)$, you will often see written as $E\left((X-\mu_X)^2\right)$.
Compute the standard deviation for each of the examples discussed so far.