RStudio projects for probability and statistics

Table of Contents

1. Probability Definition: Events, Sample Points and Sequencing Events Techniques
2. How to calculate probability: Combinations, Permutations, Cardinality
3. Random Variables: Discrete and Continuous (pmf, CDF functions)
4. [TODO Expected Value, Variance, Standard Deviation, Quartiles][#]
5. Probability Distributions: Binomial
6. TODO Probability Distributions: Geometric
7. Probability Distributions: Hypergeometric
8. Probability Distributions: Negative Binomial
9. Probability Distributions: Poisson

Probability Definition

Probability is the likelihood that an event will occur.

Events The probability of an event E is the cardinality of the event |E| divided by the cardinality of the sample space |S| (the "universe", S,) that the event is in.

$$\frac{|E|}{|S|}$$

Probability Technique: Sample Points

The Wackerly probability book is great, and describes the sample-point method for calculating probability.

Probability Technique: Sequenced Events

Another technique, after sample point technique, is sequenced events.

How to calculate probability

Counting Distinct Objects : Combinations and Permutations

Combinations: Order Doesn't Matter

$$ C = \frac{n!}{(n-r)!r!}$$

Examples: Out of the set S = {A, B, C}, a combination set would include AAA, AAB, ABC, .... etc, and ABA = BAA because order doesn't matter. When order doesn't matter, you don't need to count as many things, e.g. if AAB is equivalent to ABA, then those items count as one element of the set, not two.

Permutations: Order Matters

$$ P = \frac{n!}{(n-r)!}$$

Note that the denominator is smaller than in combinations. Permuations possibilities are much larger because order matters, so we have to count it all.

Examples: Out of the set S= {A, B, C}, a combination set would include AAA, AAB, ABC, .... etc, and ABA != BAA.

Cardinality

Cardinality is the number of elements in a Set.

Discrete Random Variables

TODO - in progress, as this coursework does NOT cover this material to satisfaction.

pmf: Probability "mass" function

A pmf measures the scalar value of a discrete variable, and a PDF (probability density function) measures the probability that a continuous random variable will have a certain range.

Note in R, the "density function," invoked via dhyper(y, r, N-r, n), this function measures a discrete random variable's scalar value, such as our hypergeometric example in R; there's a bit of oddness here, since we've used this function for discrete random variables.

Also in R, the "probability distribution function" is invoked via phyper(4, r, N-r, n).

CDF

TODO

Expected Value, Variance, Standard Deviation, Quartiles

Expected Value: TODO

Variance: TODO

Standard Deviation: TODO

Quartiles: TODO

Binomial Probability Distribution

To be continued when there is more time :) Essentially, repeated uniform experiments of a series of failures and successes, for example ${F,F,F,F,S,F,S,F...}$; the random variable for the binomial distribution counts the number of successes in each trial.

Distribution:

Using the binomial probability distribution formula, we know that for $n$ trials,

the pmf represented by:

$b(x; n, p) = {n \choose x}p^x(1-p)^{n-x}$

for $x = 0,1,2....$ (and $0$ otherwise).

Mean, Variance (TODO):

Geometric Probability Distribution

TODO - wrt Wackerly

Distribution (TODO)

Mean, Variance (TODO)

Hypergeometric Probability Distribution

Distribution:

For random sampling of sample size $n$ without replacement on a finite population of size $N$, particularly in cases where the sample size approaches the population size.

The denominator: counting the number of ways to select a subset of $n$ elements from a population of $N$, or, $N$ choose $n$ for the denominator e.g. sample space.

Then for the numerator, we think of $n$ objects, $r$ of which are red, and $N-r$ of which are black. Then, choosing $y$ objects from $r$ and then remaining $n-y$ objects from remaining $N-r$, such that by the $mn$ rule we have $mn = {r \choose y} {N-r \choose n -y}$. Putting this all together, we have:

$$p(y) = h(y; n,r,N) = \dfrac{{r \choose y}{N-r \choose n-y}}{{N \choose n}}$$

Mean, Variance (TODO):

Negative Binomial Distribution

Either counting the number of failures, or counting the $r$th trial where the first success occurs.

Distribution (TODO):

Mean, Variance (TODO):

Poisson Distribution

The Poisson probability distribution, used for rare events over a period of time, is also used to approximate the binomial distribution since the binomial distribution converges to the Poisson distribution. The Poisson distribution can approximate the binomial distribution in use cases for: large $N$, small $p$, and $\lambda = np \leq \approx 7$.

The Poisson distribution's probability function is $p(y) = \dfrac{\lambda^y}{y!}e^{-y}$, with $\mu = \lambda$, $\sigma^2 = \lambda$, and hence $\sigma = \sqrt{\lambda}$.