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http://www.psych.uiuc.edu/~jrfinley/p235/

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 seasonmean 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:

Values of X

Probability

Values of X

Probability

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:

Values of X

Probability

Value of X: X1 X2 ... Xk

Probability: p1 p2 ... pk

Expected Value (aka Expectation) of a Discrete Random Variable

• Expected Value: central tendency of the probability distribution of a random variable

€

E(X) = x ipii=1

k

∑

Expected Value (aka Expectation) of a Discrete Random Variable

• Expected Value:

€

E(X) = x ipii=1

k

∑Value of X: X1 X2 ... Xk


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

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0.000.050.100.150.200.250.300.350.400.45

0 1 2 3

X (Well-being score)

P (Probability)

Expected Value example

Value of X: X1 X2 ... Xk


Well-being score: 0 1 2 3

Probability: 0.10 0.20 0.40 0.30

E(X) = x1p1 + x2p2 + x3p3 + x4p4

Value of X: X1 X2 X3 X4

Probability: p1 p2 p3 p4

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

€

Var (X) = (x i − E(X))2 pii=1

k

∑

Expected Value & Variance: ALEKS Example

Random variables (9 new topic areas, to be completed by February 27) One random variable (7 topic areas) Classification of variables and levels of measurement Discrete versus continuous variables Discrete probability distribution: Basic Discrete probability distribution: Word problems Cumulative distribution function Expectation and variance of a random variable Rules for expectation and variance of random variables Two random variables (2 topic areas) Marginal distributions of two discrete random variables Joint distributions of dependent or independent random variables


QuickTime™ and a decompressor

are needed to see this picture.


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3 4 5 6

Value of X

P (Probability)

Xi pi

3 0.354 0.255 0.156 0.25

Xi pi xipi

3 0.354 0.255 0.156 0.25

Xi pi xipi

3 0.35 1.054 0.255 0.156 0.25

Xi pi xipi

3 0.35 1.054 0.25 15 0.156 0.25

Xi pi xipi

3 0.35 1.054 0.25 15 0.15 0.756 0.25

Xi pi xipi

3 0.35 1.054 0.25 15 0.15 0.756 0.25 1.5

E(X)= 4.3

E(X)

€

E(X) = x ipii=1

k

∑


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Value of X

P (Probability)

Xi pi xipi

3 0.35 1.054 0.25 15 0.15 0.756 0.25 1.5

E(X)= 4.3

E(X)

€

Var (X) = (x i − E(X))2 pii=1

k

∑Xi pi xipi Xi-E(X)3 0.35 1.054 0.25 15 0.15 0.756 0.25 1.5

Xi pi xipi Xi-E(X)3 0.35 1.05 -1.34 0.25 15 0.15 0.756 0.25 1.5

- =

Xi pi xipi Xi-E(X)3 0.35 1.05 -1.34 0.25 1 -0.35 0.15 0.756 0.25 1.5

Xi pi xipi Xi-E(X)3 0.35 1.05 -1.34 0.25 1 -0.35 0.15 0.75 0.76 0.25 1.5

Xi pi xipi Xi-E(X) (X i-E(X)) 2

3 0.35 1.05 -1.34 0.25 1 -0.35 0.15 0.75 0.76 0.25 1.5 1.7

2


3 0.35 1.05 -1.3 1.694 0.25 1 -0.3 0.095 0.15 0.75 0.7 0.496 0.25 1.5 1.7 2.89


3 0.35 1.05 -1.3 1.694 0.25 1 -0.3 0.095 0.15 0.75 0.7 0.496 0.25 1.5 1.7 2.89


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Value of X

P (Probability)

E(X)= 4.3

E(X)

€

Var (X) = (x i − E(X))2 pii=1

k

∑Xi pi xipi Xi-E(X) (X i-E(X)) 2 (X i-E(X)) 2pi

3 0.35 1.05 -1.3 1.694 0.25 1 -0.3 0.095 0.15 0.75 0.7 0.496 0.25 1.5 1.7 2.89

Xi pi xipi Xi-E(X) (X i-E(X)) 2 (X i-E(X)) 2pi

3 0.35 1.05 -1.3 1.69 0.5924 0.25 1 -0.3 0.095 0.15 0.75 0.7 0.496 0.25 1.5 1.7 2.89

* =Xi pi xipi Xi-E(X) (X i-E(X)) 2 (X i-E(X)) 2pi

3 0.35 1.05 -1.3 1.69 0.5924 0.25 1 -0.3 0.09 0.0235 0.15 0.75 0.7 0.49 0.0746 0.25 1.5 1.7 2.89 0.723

Var(X)= 1.41

Properties of Expectation & Variance of a Random

Var.

Expected Value of a Constant

• E(a) = a

Value of X: X1

Probability: p1

Value of a: aProbability: 1.00 a*1=a

Value of 5: 5Probability: 1.00 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

Value of X(Weight in pounds): 100 120 140 160 180

Probability: 0.10 0.15 0.30 0.25 0.20

E(X)=146 lb.

But wait!! The scale was off by 20 lb! Have to add 20 to all values...

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Value of X (weight in pounds)

P (Probability)

Adding a Constant


Probability: 0.10 0.15 0.30 0.25 0.20

E(X)=146 lb.

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P (Probability)


Probability: 0.10 0.15 0.30 0.25 0.20

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

Adding a Constant


Probability: 0.10 0.15 0.30 0.25 0.20

E(X)=146 lb.

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Value of X (weght in pounds)

P (Probability)


Probability: 0.10 0.15 0.30 0.25 0.20

E(X)= 166 lb. =146+20 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) = a2*Var(X)• How is this relevant to anything?

TRANSFORMING data.

• Ex: say you had data on peoples’ heights...


E(X)=1.7 meters

But wait!! We want height in feet! To convert, have to multiply all values by 3.28...

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1.5 1.6 1.7 1.8 1.9

Value of X (height in meters)

P (Probability)

Value of X(height in meters) 1.5 1.6 1.7 1.8 1.9

Probability: 0.10 0.25 0.30 0.25 0.10


E(X)=1.7 meters

E(X)= 5.58 ft =3.28*1.7 E(aX) = a*E(X)

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P (Probability)


Probability: 0.10 0.25 0.30 0.25 0.10

Value of X(height in feet): 4.92 5.25 5.58 5.90 6.23

Probability: 0.10 0.25 0.30 0.25 0.10


E(X)=1.7 meters

E(X)= 5.58 ft =3.28*1.7 E(aX) = a*E(X)

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P (Probability)


Probability: 0.10 0.25 0.30 0.25 0.10

Value of X(height in feet): 4.92 5.25 5.58 5.90 6.23

Probability: 0.10 0.25 0.30 0.25 0.10

Note: the whole distribution shifts to the right, AND it gets more spread out! The variance has increased!

Var(aX) = a2*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)=a2*Var(X)• Var(X2)=Var(X)+E(X)2

• If X & Y ind. Var(X+Y)=Var(X)

+Var(Y)

Var(X) = E(X2) - (E(X))2

E(X2) = Var(X) + (E(X))2


Random variables (9 new topic areas, to be completed by February 27) One random variable (7 topic areas) Classification of variables and levels of measurement Discrete versus continuous variables Discrete probability distribution: Basic Discrete probability distribution: Word problems Cumulative distribution function Expectation and variance of a random variable Rules for expectation and variance of random variables Two random variables (2 topic areas) Marginal distributions of two discrete random variables Joint distributions of dependent or independent random variables



ALEKS problem

E(X+a)E(aX)E(X+Y) algebra!

Var(aX)Var(X±a)

Var(X) = E(X2) - (E(X))2

E(X2) = Var(X) + (E(X))2

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