Probability Distributions

Bernoulli Distribution

• Defined for discrete values 0 and 1.
• Expectation: p
• Variance: p(1-p)
• PDF: Can use binomial PDF with n=1.
• CDF: Can use binomial CDF with n=1.
• Bernoulli is a Bernoulli chain of length n=1.

Binomial Distribution

• Defined on discrete values from 0 to n.
• Expectation: np
• Variance: np(1-p)
• PDF: Binomial PDF
• CDF: Binomial CDF

Uniform Distribution

• Defined on the interval [a, b].
• Expected value: (a+b)/2
• Variance: (b-a)^2 / 12
• PDF: 1 / (b-a)
• CDF: (x-a) / (b-a)
• Qth quantile: a + q(b-a)

Normal Distribution (Gaussian Distribution)

• Most important distribution
• Examples: human heights, measurement errors, molecular motion
• Density has a bell-shaped curve.
• Distribution is symmetrical (not skewed)
• Defined on all real numbers (-infinity to +infinity)
• Function value depends on x values.
• Value drops with increasing distance from the center.

Formal Notation of Normal Distribution

• Notation: N(μ, σ²) where μ is the mean and σ² is the variance.
• Expected value E[X] = μ
• Variance Var[X] = σ²
• Variance values must be positive.
• CDF has an S-shaped form.
• CDF computation involves tables, computers, or calculators.
• PDF is computed with a computer

Standard Normal Distribution

• Special case of normal distribution with mean 0 and variance 1: N(0, 1)
• Crucial importance: any normal distribution can be derived from it
• The distribution function for the N(0,1) has it's own formula Phi

Rules for Working with Standard Normal Distribution (Phi)

• Φ(-a) = 1 - Φ(a) for computing phi at negative values

Pocket Calculator Usage - Normalcdf

• Function: normalcdf (lb, ub, mu, sigma)
  • Important: Input standard deviation (sigma), not variance (sigma squared), into calculator.
• Mu and sigma define the particular normal distribution being addressed.
• lb = Lower bound
• ub = Upper bound
• For N(0, 1), mu=0 and sigma=1

Pocket Calculator Usage - Normalpdf

• Function: normalpdf (x, mu, sigma)
  • Important: Input standard deviation (sigma), not variance (sigma squared), into calculator.

Probability and Area Under the Curve

• Probability is related to the area under the curve
• Area under the curve for a threshold: The yellow area is computed with the phi functions, which is the cdf for n(0,1) distribution
• Calculating using Normalcdf: normalcdf(lower bound, upper bound, mu, sigma) because the area is equal to the cdf
• Lower bound: negative infinity -> replaced by -1EE99 in calculator
• Upper bound: threshold
• Entire Area starts and negative infinity and ends are positive infinity, area is 1.0

Relationship between Probability, Area and Calculator

• Probability that random variable x <= threshold refers to the area under the curve
• Computed using calculator's normalcdf function
• Area helps to compute the probability
• Formal Notation for Probability: express f as a function in term of Phi and each Phi is computed to get final result
• Can use pocket calculator
• Can use statistical tables

Quantiles

• CDF (Φ) takes real number as input and gives a probability
• Inverse Function (Φ^-1) starts with particular probability and results in x value
• To the power of -1 is to apply the inverse function

Using Quantiles and percentages

• If P is p = 0.75, then its the 75% function
• P = probability

Pocket Calculator for Quantiles

• Function: invNorm (area as a probability, mu, sigma)
• Calculates the x-value (quantile) for a given probability (area)

Quantiles and Standard Normal Distribution

• The x 90 percent quantile represents a threshold where 90% of the area under a curve is to the left, and 10% is to the right.
• Finding this threshold involves determining the position where 90% of the area is on the left side.
• Instead of trial and error, the formula x 90 percent = φ-1(0.9) can be used, which calculates the inverse of the standard normal cumulative distribution function at 0.9.
• Using a calculator, this is equivalent to the inverse normal function with area=0.9, μ=0, and σ=1, resulting in a value of 1.285.
• This threshold (1.285) separates 90% of the area to the left and 10% to the right in a standard normal distribution.

Completing Entries for Standard Normal Distribution

• With the knowledge of quantiles, the entries for the standard normal distribution can be completed.
• The inverse of the standard normal cumulative distribution function (φ-1) can be computed using the invNorm function on a calculator.
• For the standard normal distribution, the expected value is μ=0, and the variance is 1.
• The functions normal pdf, normal cdf, and inv normal can be used to compute the probability density function, cumulative distribution function, and quantiles, respectively.

General Normal Distribution

• The discussed standard normal distribution is (0,1).
• The general normal distribution is defined with parameters N(172, 42.25), where 172 is the mean (μ) and 42.25 is the variance (σ^2).
• This distribution represents the height distribution (stature) measured in centimeters, with the average height (mean) being 172 cm.

Probability Calculation

• The probability of a random variable (height, X) falling between 162 cm and 182 cm can be computed.
• This probability is related to the area under the normal distribution curve between these two values.
• Using the normal cdf function on a calculator with μ=172, σ=√42.25=6.5, lower bound = 162, and upper bound = 182, the resulting value is 0.876.
• The probability that a randomly selected individual has a height between 162 cm and 182 cm is 87.6%.

Impact of Variance

• The impact of variance on the normal distribution is examined using N(172, 42.25) as an example.
• The center of the distribution remains at 172 (the mean), while the variance affects the spread of the curve.
• A smaller variance (e.g., 36) results in a narrower, less spread-out curve in red.
• A larger variance (e.g., 62) results in a wider, more spread-out curve in green.
• The variance indicates the amount of variation or dispersion in the distribution.