More Probability and Introduction to Random Variables
Recap
Sampling with and without replacement
- Sampling with replacement occurs if individuals are returned to the eligible pool for each selection.
- Sampling without replacement occurs if individuals are not returned to the eligible pool for each selection.
Note that this distinction becomes unimportant with a very large population.
Mutually Exclusive vs Independent
They are not the same thing!
Disjointness is a set property, which means that there are no overlap between sets.
Independence is a probabilistic property. Disjoint sets , while independence is when one thing happens that does not impact the probability of another thing.
Random Variables
Definition
A random variable assigns a number to each outcome of a random circumstance. Equivalently, it assigns a number to each unit in a population.
Can be discrete or continuous:
- Discrete: can take one of a countable list of distinct values. Something we count.
- Continuous: can take any value in an interval or collection of intervals. Something we measure.
A distribution of a random variable describes what values the variable can take, and how often they occur.
- \(P(X = k)\): probability that the random variable is equal to k.
- \(P(X \leq k)\): the cumulative distribution function.
- A probability histogram, or stick graph.
- Just a histogram, where there are vertical lines representing probabilities instead of bars.