STATS 250

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.

Distribution

A distribution of a random variable describes what values the variable can take, and how often they occur.

Describing distribution:

  • \(P(X = k)\): 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.