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DON’T FORGET TO SIGN IN FOR CREDIT! Special Lecture: Random Variables http://www.psych.uiuc.edu/~jrfinley/p235/

Announcements • Assessment Next Week • Same procedure as last time. • AL1: Monday, Rm 289 between 9-5 • BL1: Wednesday, Rm 289 between 9-5 • Can schedule a specific time by contacting TA • Remember: Bring photo ID • Get as far through the material in ALEKS as you can before the test. You should aim to be at least halfway through the Inference slice.

Random Variables Random Variable: variable that takes on a particular numerical value based on outcome of a random experiment Random Experiment(aka Random Phenomenon): trial that will result in one of several possible outcomes can’t predict outcome of any specific trial can predict pattern in the LONG RUN that is, each possible outcome has a certain PROBABILITY of occurring

Random Variables • Examples: • # of heads in 3 coin tosses • a student’s score on the ACT • points scored by Illini basketball team in first game of the season • mean snowfall in February in Urbana • height of the next person to walk in the door

Random Variable Example & Notation • X= how many years a UIUC psych grad student takes to complete PhD • this is our random variable • xi=some particular value that X can take on • i=1 --> x1=smallest possible value of X • i=k --> xk=largest possible value of X • so for example: • x1=4 years • x2=5 years • x3=6 years • ... • xk=x7=10 years

Discrete vs. Continuous Random Variables • Discrete • Finite number of possible outcomes • ex: ACT score • Continuous • Infinitely many possible outcomes • ex: temperature in Los Angeles tomorrow • ALEKS problems: only calculating expected value and variance for DISCRETE random variables

Probability Distributions • Probability Distribution: • the possible values of a Random Variable, along with the probabilities that each outcome will occur • Graphic Depictions: • Discrete: • Continuous:

Probability Distributions Probability Distribution: the possible values of a Random Variable, along with the probabilities that each outcome will occur Graphic Depictions: Discrete: Table: Discrete:

Expected Value (aka Expectation) of a Discrete Random Variable • Expected Value: central tendency of the probability distribution of a random variable

Expected Value (aka Expectation) of a Discrete Random Variable • Expected Value: E(X) = x1p1 + x2p2 + ... + xkpk Note: the Expected Value is not necessarily a possible outcome...

Expected Value example • Say you’re given a massive set of data: • well-being scores for all senior citizens in Champaign County • possible scores: 0-3 • Random Variable: • X=Well-being score of a Champaign County senior

Expected Value example E(X) = x1p1 + x2p2 + x3p3 + x4p4 E(X) = (0)(.1) + (1)(.2) + (2)(.4) + (3)(.3) =1.9

Variance of a Discrete Random Variable • Variance (of Random Variable): measure of the spread (aka dispersion) of the probability distribution of a random variable

Expected Value & Variance: ALEKS Example

Expected Value & Variance: ALEKS Example E(X) E(X)= 4.3

- = Expected Value & Variance: ALEKS Example E(X) 2 E(X)= 4.3

* = Expected Value & Variance: ALEKS Example E(X) Var(X)= E(X)= 1.41 4.3

Properties of Expectation & Variance of a Random Var.

a*1=a 5*1=5 Expected Value of a Constant • E(a) = a

Adding a constant • E(X+a) = E(X) + a • Var(X±a) = Var(X) • How is this relevant to anything? • TRANSFORMING data. • Ex: say you had data on the initial weights of all patients in a clinical trial for a new drug to treat depression...

Adding a Constant E(X)=146 lb. But wait!! The scale was off by 20 lb! Have to add 20 to all values...

Adding a Constant E(X)=146 lb. E(X)= 166 lb. =146+20 E(X+a) = E(X) + a

Adding a Constant Note: the whole distribution shifts to the right, but it doesn’t change shape! The variance (spread) stays the same. Var(X±a) = Var(X) E(X)=146 lb. E(X)= 166 lb. =146+20 E(X+a) = E(X) + a

Multiplying by a Constant • E(aX) = a*E(X) • Var(aX) = a2*Var(X) • How is this relevant to anything? • TRANSFORMING data. • Ex: say you had data on peoples’ heights...

Multiplying by a Constant E(X)=1.7 meters But wait!! We want height in feet! To convert, have to multiply all values by 3.28...

Multiplying by a Constant E(X)=1.7 meters E(X)= 5.58 ft =3.28*1.7 E(aX) = a*E(X)

Multiplying by a Constant [Draw new distribution on chalkboard.] Note: the whole distribution shifts to the right, AND it gets more spread out! The variance has increased! Var(aX) = a2*Var(X) E(X)=1.7 meters E(X)= 5.58 ft =3.28*1.7 E(aX) = a*E(X)

Usefulness of Properties • Don’t have to transform each possible value of a random variable • Can just recalculate the expected value and variance.

Two Random Variables • E(X+Y)=E(X)+E(Y) • and if X & Y are independent: • E(X*Y)=E(X)*E(Y) • Var(X+Y)=Var(X)+Var(Y) • How is this relevant? • Difference scores (pretest-posttest) • Combining Measures

Expected Value E(a)=a E(aX)=a*E(X) E(X+a)=E(X)+a E(X+Y)=E(X)+E(Y) If X & Y ind. E(XY)=E(X)*E(Y) Variance Var(X±a) = Var(X) Var(aX)=a2*Var(X) Var(X2)=Var(X)+E(X)2 If X & Y ind. Var(X+Y)=Var(X)+Var(Y) All properties • Var(X) = E(X2) - (E(X))2 • E(X2) = Var(X) + (E(X))2

Expected Value & Variance: ALEKS Example

E(X+a) E(aX) E(X+Y) Var(aX) Var(X±a) Var(X) = E(X2) - (E(X))2 E(X2) = Var(X) + (E(X))2 ALEKS problem algebra!

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