Probability Distributions

Robert W. Walker
September 10, 2018

Probability: The Logic of Science

Jaynes presents a few core ideas and requirements for his rational system. Probability emerges as the representation of circumstances in which any given realization of a process is either TRUE or FALSE but both are possible and expressable by probabilities

  • that sum to one for all events
  • are greater than or equal to zero for any given event

General Representation of Probability

Is of necessity two-dimensional,

  • We have \( x \) and
  • we have \( Pr(X=x) \) in one of two forms (Pr or f).

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Probability Distributions of Two Forms

Our core concept is a probability distribution just as above. These come in two forms for two types [discrete (qualitative)] and continuous (quantitative)]:

  • Assumed
  • Derived

The Poster and Examples

Distributions are nouns. Sentences are incomplete without verbs – parameters. We need both; it is for this reason that the former slide is true. We do not always have a grounding for either the name or the parameter.

Continuous vs. Discrete Distributions

The differences are sums versus integrals. Why?

  • Histograms or
  • Density Plots

The probability of exactly any given value is zero on a true continuum.

Expectation

\[ E(X) = \sum_{x \in X} x \cdot Pr(X=x) \] \[ E(X) = \int_{x \in X} x \cdot f(x)dx \]

Variance

\[ E[(X-\mu)^2] = \sum_{x \in X} (x-\mu)^2 \cdot Pr(X=x) \] \[ E((X-\mu)^2) = \int_{x \in X} (x-\mu)^2 \cdot f(x)dx \]

Distributions in R

Are defined by four core parts:

  • r, for random variables
  • p, for cumulative probability [counting from left]
  • d, for density/probability that \( X=x \)
  • q, for quantile

Distributions We Deploy

  • Normal [and functions of it] [C]
  • Bernoulli and Binomial [D]
  • Geometric [D]
  • Poisson/Negative Binomial [D]
  • Uniform [C]

The Normal Distribution [Gaussian]

\[ f(x|\mu,\sigma^2 ) = \frac{1}{\sqrt{2\pi\sigma^{2}}} \exp \left[ -\frac{1}{2} \left(\frac{x - \mu}{\sigma}\right)^{2}\right] \] Is the workhorse of statistics. Key features:

  • Is self-replicating: sums of normals are normal.
  • If \( X \) is normal, then \[ Z = \frac{(X - \mu)}{\sigma} \] is normal.
  • Aside, \[ z_{x} = \frac{(x - \overline{x})}{s_{x}} \] has mean 0 and variance/std. dev. 1.

The Canonical Normal: Z

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Why Normals?

  • The Central Limit Theorem
  • They Dominate Ops [\( 6\sigma \)]
  • Normal Approximations Abound

An Example: Tire Lifetimes

A brand of steel belted radial tire has a lifetime expressed in miles that is normal with mean 96000 and standard deviation 12000. Two questions:

  1. If a warranty is set for 60,000 miles, what proportion of tires should require warranty service?

  2. What mileage requires only a 1% replacement rate?

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Tire Lifetimes

A brand of steel belted radial tire has a lifetime expressed in miles that is normal with mean 96000 and standard deviation 12000. Two questions: Normal(96000,12000)

  1. If a warranty is set for 60,000 miles, what proportion of tires should require warranty service?
pnorm(60000,96000,12000)

[1] 0.001349898

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An Example: Tires and Warranties

A brand of steel belted radial tire has a lifetime expressed in miles that is normal with mean 96000 and standard deviation 12000. Two questions: Normal(96000,12000)

  1. What mileage requires only a 1% replacement rate?
qnorm(0.01,96000,12000)

[1] 68083.83

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Bernoulli Trials and the Binomial

Suppose the variable of interest is discrete and takes only two values: yes and no. For example, is a customer satisfied with the outcomes of a given service visit?

For each individual, because the probability of yes (\( \pi \)) and no must sum to one, we can write:

\[ f(x|\pi) = \pi^{x}(1-\pi)^{1-x} \]

For multiple identical trials, we have the Binomial:

\[ f(x|n,\pi) = {n \choose k} \pi^{x}(1-\pi)^{n-x} \] where \[ {n \choose k} = \frac{n!}{(n-k)!} \]

The Binomial

BinomialR

Example: 0.5?

Post service, 100 customers were surveyed and asked one question: Satisfied? 47 said yes Does this mean we are awful?

Example: 0.5?

Post service, 100 customers were surveyed and asked one question: Satisfied? 47 said yes Does this mean we are awful? Binomial(0.5,n=100)

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Example: 0.5?

Post service, 100 customers were surveyed and asked one question: Satisfied? 47 said yes

Does this mean we are awful? Binomial(0.5,n=100)

No. If customers were indifferent [p=0.5], we should see only 47 satisfied about 30% of the time.

pbinom(47,100,0.5)

[1] 0.3086497

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Geometric Distributions

How many failures before the first success? Now defined exclusively by \( p \). In each case, (1-p) happens \( k \) times. Then, on the \( k+1^{th} \) try, p. Note 0 failures can happen…

\[ Pr(y=k) = (1-p)^{k}p \]

Example: Entrepreneurs

Suppose any startup has a \( p=0.1 \) chance of success. How many failures?

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Example: Entrepreneurs

Suppose any startup has a \( p=0.1 \) chance of success. How many failures for the average/median person?

qgeom(0.5,0.1)

[1] 6

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Events: The Poisson

Poisson

Take a binomial with \( p \) very small and let \( n \rightarrow \infty \). We get the Poisson distribution (\( y \)) given an arrival rate \( \lambda \) specified in events per period.

\[ f(y|\lambda) = \frac{\lambda^{y}e^{-\lambda}}{y!} \]

Examples: The Poisson

  • Walk in customers
  • Emergency Room Arrivals
  • Births, deaths, marriages
  • Prussian Cavalry Deaths by Horse Kick
  • Fish?

Emergency Room Arrivals

Suppose trauma victims arrive at a rate of 12 per hour. You schedule ER teams that can see 2 patients per hour and you want enough teams so that 95% of hours are properly staffed.

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Emergency Room Arrivals

Suppose trauma victims arrive at a rate of 12 per hour. You schedule ER teams that can see 3 patients per hour and you want enough teams so that 95% of hours are properly staffed.

qpois(0.95, 12)

[1] 18

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Deaths by Horse Kick

library(vcd)
data(VonBort)
head(VonBort)

  deaths year corps fisher
1      0 1875     G     no
2      0 1875     I     no
3      0 1875    II    yes
4      0 1875   III    yes
5      0 1875    IV    yes
6      0 1875     V    yes

mean(VonBort$deaths)

[1] 0.7

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The Uniform Distribution

  • Is flat, each value is eqully likely.
  • Defined on 0 to 1 gives a random cumulative probability \( X \leq x \).

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Monte Carlo Simulation

Putting it together.

  1. Customers arriving at a car dealership at a rate of 6 per hour.
  2. Each customer has a 15% probability of making a purchase.
  3. Purchases have a. uniform profits over the interval $1000-$3000. b. Normal profits that average 1500 with standard deviation 500

Let's produce and graph the solution.

Monte Carlo Simulation

Putting it together. r, let's try 1000.

  1. Customers arriving at a car dealership at a rate of 6 per hour.
  2. Each customer has a 15% probability of making a purchase.
  3. Purchases have
    • Uniform profits over the interval $1000-$3000.
    • Normal profits that average $1500 with standard deviation $500 * Let's produce and graph the solution. ***
Customers <- rpois(1000, 6)
Purchasers <- rbinom(1000, size=Customers, prob=0.15)
Profits.U <- sapply(c(1:1000), function(x) { sum(runif(Purchasers[[x]], 1000, 3000))} )
Profits.N <- sapply(c(1:1000), function(x) { sum(rnorm(Purchasers[[x]], 1500, 500))} )

Solutions

par(mfrow=c(1,1))
plot(x=Customers, y=Purchasers)

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Your First Quiz Module

Will play off of Monte Carlo Simulation such as this.