Random Variables and Probability Distributions

Random Variables and Probability Distributions • Random Variables - Random outcomes corresponding to subjects randomly selected from a population. • Probability Distributions - A listing of the possible outcomes and their probabilities (discrete r.v.s) or their densities (continuous r.v.s) • Normal Distribution - Bell-shaped continuous distribution widely used in statistical inference • Sampling Distributions - Distributions corresponding to sample statistics (such as mean and proportion) computed from random samples

Discrete Probability Distributions • Discrete RV - Random variable that can take on a finite (or countably infinite) set of discontinuous possible outcomes (Y) • Discrete Probability Distribution - Listing of outcomes and their corresponding probabilities (y , P(y))

Example - Supreme Court Vacancies • Supreme Court Vacancies by Year 1837-1975 • Y  # Vacancies in Randomly selected year Source: R.J. Morrison (1977), “FDR and the Supreme Court: An Example of the Use of Probability Theory in Political History”, History and Theory, Vol. 16, pp 137-146

Parameters of a P.D. • Mean (aka Expected Value) - Long run average outcome • Standard Deviation - Measure of the “typical” distance of an outcome from the mean

Example - Supreme Court Vacancies

Normal Distribution • Bell-shaped, symmetric family of distributions • Classified by 2 parameters: Mean (m) and standard deviation (s). These represent location and spread • Random variables that are approximately normal have the following properties wrt individual measurements: • Approximately half (50%) fall above (and below) mean • Approximately 68% fall within 1 standard deviation of mean • Approximately 95% fall within 2 standard deviations of mean • Virtually all fall within 3 standard deviations of mean • Notation when Y is normally distributed with mean m and standard deviation s :

Normal Distribution

Example - Heights of U.S. Adults • Female and Male adult heights are well approximated by normal distributions: YF~N(63.7,2.5) YM~N(69.1,2.6) Source: Statistical Abstract of the U.S. (1992)

Standard Normal (Z) Distribution • Problem: Unlimited number of possible normal distributions (- < m <  , s > 0) • Solution: Standardize the random variable to have mean 0 and standard deviation 1 • Probabilities of certain ranges of values and specific percentiles of interest can be obtained through the standard normal (Z) distribution

Standard Normal (Z) Distribution • Standard Normal Distribution Characteristics: • P(Z  0) = P(Y  m ) = 0.5000 • P(-1  Z  1) = P(m-s Y m+s ) = 0.6826 • P(-2  Z  2) = P(m-2s Y m+2s ) = 0.9544 • P(Z  za) = P(Z  -za) = a (using Z-table)

Finding Probabilities of Specific Ranges • Step 1 - Identify the normal distribution of interest (e.g. its mean (m) and standard deviation (s) ) • Step 2 - Identify the range of values that you wish to determine the probability of observing (YL , YU), where often the upper or lower bounds are  or - • Step 3 - Transform YL and YU into Z-values: • Step 4 - Obtain P(ZL Z  ZU) from Z-table

Example - Adult Female Heights • What is the probability a randomly selected female is 5’10” or taller (70 inches)? • Step 1 -Y ~ N(63.7 , 2.5) • Step 2 -YL = 70.0 YU =  • Step 3 - • Step 4 - P(Y  70) = P(Z  2.52) = .0059 (  1/170)

Finding Percentiles of a Distribution • Step 1 - Identify the normal distribution of interest (e.g. its mean (m) and standard deviation (s) ) • Step 2- Determine the percentile of interest 100p% (e.g. the 90th percentile is the cut-off where only 90% of scores are below and 10% are above) • Step 3 - Turn the percentile of interest into a tail probability a and corresponding z-value (zp): • If 100p 50 then a = 1-p and zp = za • If 100p< 50 then a = p and zp = -za • Step 4 - Transform zp back to original units:

Example - Adult Male Heights • Above what height do the tallest 5% of males lie above? • Step 1 - Y ~ N(69.1 , 2.6) • Step 2 - Want to determine 95th percentile (p = .95) • Step 3 - Since 100p > 50, a = 1-p = 0.05 zp = za = z.05 = 1.645 • Step 4 - Y.95 = 69.1 + (1.645)(2.6) = 73.4

Statistical Models • When making statistical inference it is useful to write random variables in terms of model parameters and random errors • Here m is a fixed constant and e is a random variable • In practice m will be unknown, and we will use sample data to estimate or make statements regarding its value

Sampling Distributions and the Central Limit Theorem • Sample statistics based on random samples are also random variables and have sampling distributions that are probability distributions for the statistic (outcomes that would vary across samples) • When samples are large and measurements independent then many estimators have normal sampling distributions (CLT): • Sample Mean: • Sample Proportion:

Example - Adult Female Heights • Random samples of n = 100 females to be selected • For each sample, the sample mean is computed • Sampling distribution: • Note that approximately 95% of all possible random samples of 100 females will have sample means between 63.0 and 64.0 inches

Random Variables and Probability Distributions