DON’T FORGET TO SIGN IN FOR CREDIT! Special Lecture: Random Variables

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 x i =some particular value that X can take on  i=1 --> x 1 =smallest possible value of X  i=k --> x k =largest possible value of X so for example:  x 1 =4 years  x 2 =5 years  x 3 =6 years ...  x k =x 7 =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) = x 1 p 1 + x 2 p x k p k 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) = x 1 p 1 + x 2 p 2 + x 3 p 3 + x 4 p 4 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

E(X)= 4.3 E(X)

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

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

Properties of Expectation & Variance of a Random Var.

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

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.= E(X+a) = E(X) + a

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

Multiplying by a Constant E(aX) = a*E(X) Var(aX) = a 2 *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

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 E(X)=1.7 meters E(X)=5.58 ft=3.28*1.7 E(aX) = a*E(X) Note: the whole distribution shifts to the right, AND it gets more spread out! The variance has increased! Var(aX) = a 2 *Var(X) [Draw new distribution on chalkboard.]

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

All properties 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)=a 2 *Var(X) Var(X 2 )=Var(X)+E(X) 2 If X & Y ind.  Var(X+Y)=Var(X)+Var(Y) Var(X) = E(X 2 ) - (E(X)) 2 E(X 2 ) = Var(X) + (E(X)) 2

Expected Value & Variance: ALEKS Example

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