##### Contents

- Specifying Probabilities
- Probability Distributions
- Random Variables
- Binomial Distribution
- Normal Distribution
- Expectation of a Random Variable
- Standard Deviation of a Random Variable
- Variance of a Random Variable

Inferential Statistics uses random variables and probability distributions.

##### 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.

- The

- 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

- A

- 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

- the binomial distribution:

- 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

- 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 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

- 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 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:

- 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 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:

- 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 more intuitive, but mathematically recalcitrant, average is the
- 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}

- That is, E(X – E(X))

- 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

- The variance of X = the expectation of (X – the expectation of X)

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

- E(X – E(X))

- Alternatively
- E(X
^{2}) – (E(X))^{2}

- E(X