Random Variables and Probability Distributions

Table of Contents

Specifying Probabilities

Single Probability
- The probability of rolling a seven = ⅙
Probability Range
- The probability of rolling 5 through 9 = ⅔
- The probability of rolling at least 10 = ⅙
Probability Distribution
- The probabilities of rolling 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 are 1/36, 1/18, 1/12, 1/9, 5/36, 1/6, 5/36, 1/9, 1/12, 1/18, 1/36 respectively.

Probability Distribution

A probability distribution is an assignment of probabilities to a set of values.
- The Rolling Dice distribution assigns probabilities to the possible outcomes of a roll, the integers 2 through 12.

Probability distributions
- have names like
  - Normal, Poisson, Chi-Square, Student T, Binomial, Hypergeometric, Beta, F
- are discrete or continuous
  - A discrete distribution has gaps between its values
  - A continuous distribution has no gaps between its values, e.g. the normal distribution
  - Rolling Dice is discrete
- may take input parameters, e.g.
  - the binomial distribution: number of trials and probability of success
  - the normal distribution: mean and standard deviation
  - Rolling Dice has no parameters
- have properties such as
  - expectation (mean), median, standard deviation, variance, quantile, skewness, kurtosis
  - Rolling Dice
    - Expectation = 7
    - Median = 7
    - Standard deviation = 2.41523
    - 1/4 Quantile = 5
- sum to 1
  - Rolling Dice
    - 1/36 + 1/18 + 1/12 + 1/9 + 5/36 + 1/6 + 5/36 + 1/9 + 1/12 + 1/18 + 1/36 = 1

Individual probabilities and ranges can be calculated from a probability distribution
- Boxcars
  - Probability[x = 12, Distributed[x, Dice]] = 1/36
- 7 or 11
  - Probability[x = 7 or x = 11, Distributed[x, Dice]] 2/9
- x ≥ 10
  - Probability[x ≥ 10, Distributed[x, Dice]] = 1/6

A probability distribution is different from a frequency distribution.

View Interactive on Mean, Median, and Quartiles
- Mathematica Demonstration by Ian McLeod

Random Variables

Random variables are variables with built-in probability distributions.
Let random variable D be the outcome of rolling a pair of dice. Then:

A random variable
- is usually designated by a capital letter
- is defined by its probability distribution
- has the properties of its probability distribution
  - discrete or continuous
  - expectation (mean), median, standard deviation, variance, quantile, skewness, kurtosis

Random variables are a neat, simple way of doing math involving probability distributions. For example:
- Let random variable Y = BinomialDistribution[10, 0.2].
- Let random variable Z = BinomialDistribution[10, 0.8]
  - Then YZ = BinomialDistribution[10, 0.2] x BinomialDistribution[10, 0.8]
- Thus,
  - If P[Y=4] = 0.0880804 and P[Z=4] = 0.00550502, then
    - P[YZ=4] = 0.0880804 x 0.00550502 = 0.00048488

Binomial Distribution

The binomial distribution gives the probabilities of the possible outcomes of a series of binary tests, called Bernoulli Trials, where the result is either one thing or another: success or failure, heads or tails, right or wrong, 1 or 0.
It has two input parameters:
- n = the number of trials.
- p = the probability of success (heads, right, 1) on a given trial.

Let random variable X = the number of heads in 10 flips of an unbiased coin
- The values of X are 0 through 10 and assigned probabilities by the Binomial Distribution[n,p], where n = 10 and p = 0.5
That is, X ~ Binomial Distribution[10, 0.5]
- where the tilde means “has probability distribution such and such”

This bar chart displays the probabilities assigned to the values of X.
- The values of X are on the horizontal axis
- The probabilities of X are on the vertical axis.

The binomial distribution for different n and p:

Increasing n moves the distribution to the right

Changing p moves the distribution and alters its shape

The binomial distribution is discrete, meaning there are gaps between values.
- P[X = 4] = 0.205078
- P[X = 4.5] is undefined
- P[X = 5] = 0. 246093
  - where X ~ Binomial Distribution[10, 0.5]

The probabilities of the values of a random variable total one.
- Probability[x ≥0 and x ≤10, BinomialDistribution[10,0.5]] = 1.0

Examples of probabilities derived from the binomial distribution, using Mathematica
- Probability[x ≥ 0 , BinomialDistribution[10, 0.5]] = 1
- Probability[x =5 , BinomialDistribution[10, 0.5]] = 0.246094
- Probability[x ≥ 8 , BinomialDistribution[10, 0.5]] = 0.0546875
- Probability[x ≥ 4 and x ≤ 6 , BinomialDistribution[10, 0.5]] = 0.65625
- Probability[x ≥ 8 or x ≤ 2 , BinomialDistribution[10, 0.5]] = 0.109375

Random numbers generated by the binomial distribution, using Mathematica
- RandomVariate[BinomialDistribution[10, 0.5], 50]
  - 6,5,5,6,4,5,4,3,7,5,3,4,8,5,5,6,4,7,0,5,6,5,7,5,4,4,5,3,8,3,3,5,6,4,7,6,6,9,3,5,5,7,2,5,6,4,5,3,4,5

Normal Distribution

The most widely used continuous distribution is the Normal Distribution, a bell-shaped, symmetric distribution that approximates natural quantities such as blood pressure, income, and measurement errors.
The Normal Distribution has two input parameters:
- μ = the mean of the distribution
- σ = the standard deviation of the distribution
A normal distribution that approximates adult male heights in inches:

The standard normal distribution is defined by its parameters: μ = 0 and σ = 1:

The normal distribution for other μ and σ:

Increasing σ flattens the distribution

Changing μ moves the distribution left and right

The normal distribution is continuous, meaning there are no gaps between values.
- Thus:

Examples of probabilities derived from the normal distribution for μ=70 and σ = 4, using Mathematica
- Probability[x ≥ (12 x 6), NormalDistribution[70, 4]] = 0.31
- Probability[x ≥ (12 x 6.5), NormalDistribution[70, 4]] = 0.023
- Probability[x ≥ (12 x 5) and x ≤ (12 x 6), NormalDistribution[70, 4]] = 0.69

The probability between two numbers under a continuous distribution is defined as the area under its curve between the numbers.
- The probability that a random number falls between 0.4 and 0.6 under the standard normal distribution = 0.0703251

Expectation of a Random Variable

The expectation of a random variable is its probability-weighted average, i.e. the sum (or integral) of its probability-weighted values.

Expectation of the discrete random variable X, where X ~ Binomial Distribution[10, 0.5]
- Equals the sum of:

Expectation of the continuous random variable X, where X ~ NormalDistribution[5, 2]

Standard Deviation of a Random Variable

The standard deviation of a random variable is the “average” probability-weighted, mathematically-behaved distance of its values from its expectation.
- (The more intuitive, but mathematically recalcitrant, average is the mean absolute deviation.)
The standard deviation is defined as the square root of the variance.

Variance of a Random Variable

The variance of a random variable X is the expectation of (X – the expectation of X)²
- That is, E(X – E(X))²
- Alternatively, E(X²) – (E(X))²

Variance of the discrete random variable X, where X ~ Binomial Distribution[10, 0.5]
- The variance of X = the expectation of (X – the expectation of X)²
- The expectation of X = 5
- So the variance is the expectation of (X – 5)²
- Which is the probability-weighted sum of (X – 5)²
- In mathematical terms:
- Which is the sum of:
  - - = 2.5

Variance of the continuous random variable X, where X ~ continuous NormalDistribution[5, 2]
- E(X – E(X))²

Alternatively
- E(X²) – (E(X))²

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