Table of Contents
 Specifying Probabilities
 Probability Distribution
 Random Variables
 Binomial Distribution
 Normal Distribution
 Expectation of a Random Variable
 Standard Deviation of a Random Variable
 Variance of a Random Variable
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, ChiSquare, 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
 Rolling Dice
 have names like
 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
 Boxcars
 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 builtin 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
 If P[Y=4] = 0.0880804 and P[Z=4] = 0.00550502, then
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
 RandomVariate[BinomialDistribution[10, 0.5], 50]
Normal Distribution
 The most widely used continuous distribution is the Normal Distribution, a bellshaped, 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:
 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
 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 probabilityweighted average, i.e. the sum (or integral) of its probabilityweighted values.
 Expectation of the discrete random variable X, where X ~ Binomial Distribution[10, 0.5]
 Equals the sum of:
 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” probabilityweighted, mathematicallybehaved 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(X^{2}) – (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 probabilityweighted 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(X^{2}) – (E(X))^{2}