Discrete Random Variables
Discrete random variables represent quantities that can take on only specific, separate values, typically whole numbers. Discrete random variables have a countable set of possible outcomes.
Probability Distributions
The probability distribution of a discrete random variable describes the likelihood of each possible outcome. This can be represented in two main ways:
- Probability Mass Function (PMF): A function that gives the probability of each possible value.
- Cumulative Distribution Function (CDF): A function that gives the probability of the variable being less than or equal to a given value.
The sum of all probabilities in a discrete probability distribution must equal 1.
Expected Value (Mean)
The expected value, often denoted as E(X), represents the average outcome if the random process were repeated many times.
For a discrete random variable X with possible values $x_1, x_2, ..., x_n$ and corresponding probabilities $p_1, p_2, ..., p_n$, the expected value is calculated as:
$$E(X) = \sum_{i=1}^n x_i p_i$$
ExampleFor the die roll example:
$E(X) = 1(\frac{1}{6}) + 2(\frac{1}{6}) + 3(\frac{1}{6}) + 4(\frac{1}{6}) + 5(\frac{1}{6}) + 6(\frac{1}{6}) = 3.5$
The expected value of 3.5 indicates that, on average, if you rolled the die many times, the mean of all rolls would approach 3.5.
TipWhen calculating expected values, pay attention to the units. The expected value will have the same units as the random variable itself.
