1. 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

2. 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))

3. Example - Supreme Court Vacancies
Supreme Court Vacancies by Year
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

4. 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

5. Example - Supreme Court Vacancies

6. 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 :

7. Normal Distribution

8. 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)

9. 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

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

11. 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

12. 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 = YU = 
Step 3 -
Step 4 - P(Y  70) = P(Z  2.52) = (  1/170)

13. 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:

14. 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 = (1.645)(2.6) = 73.4

15. 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

16. 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:

17. 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