Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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)) 0 P( y ) 1 P ( y ) 1 all y Example - Supreme Court Vacancies • Supreme Court Vacancies by Year 18371975 • Y # Vacancies in Randomly selected year # Vacancies (y) 0 1 2 3 >3 Total Frequency (# of Years) 81 43 14 1 0 139 Proportion (P(y)) 81/139=.5827 43/139=.3094 14/139=.1007 1/139=.0072 0/139=.0000 1.0000 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 E(Y ) yP( y) Deviation - Measure of the “typical” distance of an outcome from the mean Standard E (Y ) 2 2 ( y ) P( y ) 2 2 y P ( y ) Example - Supreme Court Vacancies y P(y) yP(y) y2P(y) 0 .5827 .0000 .0000 1 .3094 .3094 .3094 2 .1007 .2014 .4028 3 .0072 .0216 .0648 Total 1.0000 .5324 .7770 yP( y) .5324 2 2 2 y P ( y ) . 7770 (. 5324 ) .4936 .7025 Normal Distribution • Bell-shaped, symmetric family of distributions • Classified by 2 parameters: Mean () and standard deviation (). 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 and standard deviation : Y ~ N ( , ) Normal Distribution P(Y ) 0.50 P( Y ) 0.68 P( 2 Y 2 ) 0.95 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) 20 20 18 16 14 12 10 10 8 6 4 Std. Dev = 2.48 Std. Dev = 2.61 2 Mean = 63.7 Mean = 69.1 0 N = 99.68 55.5 57.5 56.5 59.5 58.5 61.5 60.5 63.5 62.5 65.5 64.5 67.5 66.5 INCHESF 69.5 68.5 70.5 N = 99.23 0 59.5 61.5 63.5 65.5 67.5 69.5 71.5 73.5 75.5 60.5 62.5 64.5 66.5 68.5 70.5 72.5 74.5 76.5 INCHESM Cases weighted by PCTM Cases weighted by PCTF Source: Statistical Abstract of the U.S. (1992) Standard Normal (Z) Distribution • Problem: Unlimited number of possible normal distributions (- < < , > 0) • Solution: Standardize the random variable to have mean 0 and standard deviation 1 Y ~ N ( , ) Z Y ~ N (0,1) • 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: – – – – a za P(Z 0) = P(Y ) = 0.5000 P(-1 Z 1) = P(- Y + ) = 0.6826 P(-2 Z 2) = P(-2 Y +2 ) = 0.9544 P(Z za) = P(Z -za) = a (using Z-table) 0.500 0.000 0.100 1.282 0.050 1.645 0.025 1.960 0.010 2.326 0.005 2.576 Finding Probabilities of Specific Ranges • Step 1 - Identify the normal distribution of interest (e.g. its mean () and standard deviation () ) • 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: ZL YL ZU YU • 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 70.0 63.7 ZL 2.52 ZU 2.5 • Step 4 - P(Y 70) = P(Z 2.52) = .0059 ( 1/170) z 2.4 2.5 2.6 .00 .0082 .0062 .0047 .01 .0080 .0060 .0045 .02 .0078 .0059 .0044 .03 .0075 .0057 .0043 Finding Percentiles of a Distribution • Step 1 - Identify the normal distribution of interest (e.g. its mean () and standard deviation () ) • 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: Yp z p 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 z 1.5 1.6 1.7 .03 .0630 .0516 .0418 .04 .0618 .0505 .0409 .05 .0606 .0495 .0401 .06 .0594 .0485 .0392 Statistical Models • When making statistical inference it is useful to write random variables in terms of model parameters and random errors Y (Y ) Y • Here is a fixed constant and is a random variable • In practice 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): Y ~ N , n – Sample Mean: – Sample Proportion: (1 ) ~ N , n ^ Example - Adult Female Heights • Random samples of n = 100 females to be selected • For each sample, the sample mean is computed • Sampling distribution: 2.5 Y ~ N 63.5, N (63.5,0.25) 100 • Note that approximately 95% of all possible random samples of 100 females will have sample means between 63.0 and 64.0 inches