In jrminter/statshelpR: Convenience functions for statistical analysis

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Definition of a probability distribution

This example is adapted from 9:30 min on from the OpenIntro YouTube video.

A probability distribution is a list of all possible disjoint outcomes (or mutually-exclusive outcomes) and their probabilities.

A probability distribution must satisfy three different rules:

1. The outcomes must be disjoint.
2. Each probability must be between $0$ and $1$.
3. The total of the probabilities must equal $1$.

Consider some hypothetical distributions of income (in $1000) in the first 4 columns of the table below and evaluate them by our three rules.

+---+----+-----+--------+----+----+--+--+--+ | |0-25|25-50| 50-100 |100+|Sum |R1|R2|R3| +===+====+=====+========+====+====+==+==+==+ |(a)|0.18| 0.39| 0.33|0.16|1.06| 1| 1| 0| +---+----+-----+--------+----+----+--+--+--+ |(b)|0.38|-0.27| 0.52|0.37|1.00| 1| 0| 1| +---+----+-----+--------+----+----+--+--+--+ |(c)|0.28| 0.27| 0.29|0.16|1.00| 1| 1| 1| +---+----+-----+--------+----+----+--+--+--+

Only C meets all three criteria.

A complement

Think of these as left-overs.

$1-P(A') = P(A)$

It is useful when it is easier to compute $P(A')$ than $P(A)$.

Going back to our distribution (c), $P(0-100) = 1-P(100+) = 0.84$

Independence

Two processes are independent if knowing about one provides no useful information about the second.

In general, rolling a die or flipping a coin are independent.

So we can use the multiplication rule for independent processes.

For independent processes A and B,

$P(A\ and\ B) = P(A) \times P(B)$

Recall our General Addition Rule

$P(A\ or\ B) = P(A) + P(B) - P(A\ and\ B)$

jrminter/statshelpR documentation built on May 2, 2020, 12:08 a.m.