Statistics for Social and Behavioral Sciences Session #11: Random Variable, Expectations (Agresti and Finlay, Chapter 4) Prof. Amine Ouazad

Statistics Course Outline P ART I. I NTRODUCTION AND R ESEARCH D ESIGN P ART II. D ESCRIBING DATA P ART III. D RAWING CONCLUSIONS FROM DATA : I NFERENTIAL S TATISTICS P ART IV. : C ORRELATION AND C AUSATION : R EGRESSION A NALYSIS Week 1 Weeks 2-4 Weeks 5-9 Weeks This is where we talk about Zmapp and Ebola! Firenze or Lebanese Express?

Last Session Four rules of probability distributions 1.P(not A) = 1 – P(A) 2.P(A or B) = P(A) + P(B) when P(A and B)=0 3.P(A and B)=P(A) P(B given A) Beware of the inverse probability fallacy, P(B given A) is not P(A given B) 3’. P(A and B)=P(A) P(B) when A and B are independent Inverse Probability Fallacy: – P(A|B) is not P(B|A). – We have a formula P(A|B) = (P(B|A) P(A)) / P(B)

Outline 1.Random Variable Probability distribution of a random variable Expectation of a random variable 1.The normal distribution 2.Polls and normal distributions Next time:Probability Distributions (continued) Chapter 4 of A&F

Random variable A random variable is a variable whose value is not given ex- ante… but rather can take multiple values ex-post. Example: – X is a random variable that, before the coin is tossed (ex-ante), can take values « Heads » or « Tails ». Once the coin is tossed (ex-post), the value of X is known, it is either « Heads » or « Tails ». – Y is a random variable that can take values 1,2,3,4,5, or 6 depending on the draw of a dice. Before the dice is thrown, the value is not known. After the dice is drawn, we know the value of Y.

Probability distribution of a random variable Take all possible values of a random variable Y: – Example: 1,2,3,4,5,6 – In general: y 1, y 2, y 3, …, y K. Probability of the event that the random variable Y equates y k is noted P(Y=y k ) or simply P(y k ). The probability distribution of random variable Y is the list of all values of P(Y=y k ). Example: for a balanced dice, the probability distribution of Y is the list of values P(Y=1), P(Y=2), P(Y=3), … which is {1/6,1/6,1/6,1/6,1/6,1/6} All throughout the course we consider either discrete quantitative random variables or categorical random variables.

Expected value of a random variable What are your expected gains when playing the coin game? Gain is a random variable, equal to +10 AED when getting heads, and -10 AED when getting tails. E(gain) = Gain when getting heads x Probability of heads + Gain when getting tails x Probability of tails. In general, for a random variable Y, the expected value of Y is: E(Y) =  y k P(Y=y k ) Also note that probabilities sum to one.  P(Y=y k ) = 1 Should I play this game at all? What is my expected gain?? Should I play this game at all? What is my expected gain??

Expected Earnings? « Your annual earnings right after NYU Abu Dhabi » is a random variable… – The variable has not been realized yet. Let’s give it a name Y = « Your annual earnings right after NYU Abu Dhabi ». E(earnings) = E(Y) =  y k P(Y=y k ) Takes potentially K values. Problemo: We don’t observe earnings in the future!!! Hum, how much will I earn??

An approximation is to use the distribution of current graduates … To substitute for our lack of knowledge of P(Y=y k ) for each k. Earnings take K distinct values, no two graduates earn exactly the same annual wage… Hence an approximation of expected earnings is E(Y) =  y k x (1/ K) The average earnings of current graduates… But that’s only an approximation !! What could be wrong? Expected Earnings? Hum, how much will I earn??

Properties of the Expectation The expectation of the sum is the sum of the expectations: E(earnings – debt) = E(earnings) – E(debt) The expectation of a constant x the random variable is the constant x the expectation: E( Constant x Earnings ) = Constant x E(Earnings) E.g. E(Earnings in AED) = 3.6 x E(Earnings in USD) Beware !!! E( X Y ) is not E(X) E(Y) in general. When X and Y are independent, E( X Y ) = E(X) E(Y). Law of conditional expectation E(X)=E(E(X|Z))

Outline 1.Random Variable Probability distribution of a random variable Expectation of a random variable 1.The normal distribution 2.Polls and normal distributions Next time:Probability Distributions (continued) Chapter 4 of A&F

A particular distribution Some random variables have a particular “bell-shaped” distribution: – Individuals’ height. What is the distribution of height at age 20? P(height) What height can I expect for my child? E(height) – Individuals’ weight. What is the distribution of weight at age 35? P(weight) What weight can I expect at age 35? E(height) – The logarithm of income. What is the distribution of the log of income after graduation? P(log(income)) What log income can I expect after graduation? The “bell-shaped” distribution will now be called a “normal” distribution.

The normal distribution “The normal distribution is symmetric, bell shaped, and characterized by its mean m and standard deviation s. The probability within any particular number of standard deviations of m is the same for all normal distributions.” P(  –  < height <  +  ) = 0.68or 68% P(  - 2  < height <  + 2  ) = 0.95or 95% P(  - 3  < height <  + 3  ) = 0.997or 99.7% All of these are “events”

Draw a histogram will a very small bin size… so that the little stairs disappear…. and a curve appears. The normal distribution

Comparing test scores across colleges Test scores have a normal distribution with mean 3 and standard deviation 4. Test scores have a normal distribution with mean 4 and standard deviation 1. “Hip hop in the Middle East” “Early paleontology in Indianapolis” Problem: how do I compare Marina’s test score of 3.6 at the paleontology course with a test score of 4.1 at the Hip Hop in the Middle East?

Z-score ! Take a student’s paleontology test score at the end of the semester. This is a random variable. – Its probability distribution has a mean of  =3 with a standard deviation of  =4. – Now consider the “z-scored” paleontology test score: – The z-scored paleontology test score has a mean of 0, and a standard deviation of 1.

Standard Normal Distribution Is simply the normal distribution with mean 0 and standard deviation 1. A z-score of 3 means that the student is three times the standard deviation (of original test scores) above the mean. So who has a better grade, Marina or Slavoj?

Outline 1.Random Variable Probability distribution of a random variable Expectation of a random variable 1.The normal distribution 2.Polls and normal distributions Next time:Probability Distributions (continued) Chapter 4 of A&F

Who will win the mid term elections in the US? Mid term elections are held two years after the presidential elections in the United States. They take place early november A question: what fraction of the voters will vote for a democrat in Colorado?

Wrap up A random variable is a variable whose value has not been realized. The expectation of a random variable Y is: E(Y) =  y k P(Y=y k ) Also, E(X+Y) = E(X) + E(Y), and E(c X)=c E(X), and E(E(X|Z))=E(X) Typically the probability distribution P is not known, but we approximate it…. – Using the distribution for past values of Y (example: earnings of previous graduates) – Using polls, to ask individuals for example how they will vote. The normal distribution is an ubiquitous distribution, that is symmetric, bell shaped. It is characterized by its mean  and its standard deviation . The standard normal distribution has mean 0 and standard deviation 1.

Coming up: Readings: Chapter 4 entirely – full of interesting examples and super relevant. Online quiz tonight. Go to the website l-3845.html and prepare one or two slides to present the race in Colorado. l-3845.html – Who do you think will win? – What is MoE? – What is the likely distribution of the “fraction of voters who will vote for Gardner?” For help: Amine Ouazad Office 1135, Social Science building Office hour: Tuesday from 5 to 6.30pm. GAF: Irene Paneda Sunday recitations. At the Academic Resource Center, Monday from 2 to 4pm.