Special Lecture: Random Variables
http://www.psych.uiuc.edu/~jrfinley/p235/
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Announcements
• Assessment Next Week
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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
 Discrete:
Probability
• Graphic Depictions:
Values of X
Probability
 Continuous:
Values of X
Probability Distributions
• Probability Distribution:
 the possible values of a Random Variable, along with
the probabilities that each outcome will occur
• Graphic Depictions:
Probability
 Discrete:
Values of X
• Table:
 Discrete:
Value of X:
Probability:
X1
p1
X2
p2
...
...
Xk
pk
Expected Value (aka Expectation) of a
Discrete Random Variable
• Expected Value: central tendency of the
probability distribution of a random
variable
k
E(X)   x i pi
i1
Expected Value (aka Expectation) of a
Discrete Random Variable
• Expected Value:
k
E(X)   x i pi
i1
Value of X:
Probability:

X1
p1
X2
p2
...
...
Xk
pk
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
P (Probability)
Expected Value example
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0.40
0.30
0.20
0.10
0
1
2
3
X (Well-being score)
Value of X:
Probability:
X1
p1
X2
p2
X3
...
...
p3
X4k
p4k
Well-being
score:
Probability:
0
0.10
1
0.20
2
0.40
3
0.30
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
k
Var(X)   (x i  E(X)) pi
2
i1
Expected Value & Variance:
ALEKS Example
Rando m varia bles (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
Cumu lative distribution function
Exp ectation and variance of a random variable
Rules for expectation and variance of random variables
Two random va riables (2 topic areas)
Marginal distributions of two discrete random variables
Joint distributions of dependent or independent random variables
Expected Value & Variance:
ALEKS Example
QuickTime™ and a
decompressor
are needed to see this picture.
Expected Value & Variance: ALEKS Example
P (Probability)
0.40
0.35
0.35
0.30
0.25
0.25
0.25
0.20
0.15
0.10
0.15
0.05
0.00
E(X)
3
4
5
6
Value of X
Xi
3
4
5
6
pi
0.35
0.25
0.15
0.25
xipi
1.05
1
0.75
1.5
E(X)= 4.3
k
E(X)   x i pi
i1
Expected Value & Variance: ALEKS Example
P (Probability)
0.40
0.35
0.35
0.30
0.25
0.25
0.25
0.20
0.15
0.10
0.15
0.05
0.00
E(X)
3
4
5
Value of X
6
k
Var(X)   (x i  E(X)) 2 pi
i1
X
Xii
33
44
55
66
ppii
0.35
0.35
0.25
0.25
0.15
0.15
0.25
0.25
xxiippii
X
Xii-E(X)
-E(X) (X
(Xii-E(X))
-E(X))22
1.05
-1.3
1.05
-1.3 2
1.69
11
-0.3
-0.3
0.09
0.75
0.7
0.75
0.7
0.49

1.5
1.5 = 1.7
1.7
2.89
E(X)= 4.3
Expected Value & Variance: ALEKS Example
P (Probability)
0.40
0.35
0.35
0.30
0.25
0.25
0.25
0.20
0.15
0.10
0.15
0.05
0.00
E(X)
3
4
5
Value of X
6
k
Var(X)   (x i  E(X)) 2 pi
i1
Xi
3
4
5
6
pi
0.35
0.25
0.15
0.25
xipi
Xi-E(X) (Xi-E(X))2 (Xi-E(X))2pi
1.05 * -1.3
1.69 =
0.592
1
-0.3
0.09
0.023
0.75
0.7
0.49
0.074

1.5
1.7
2.89
0.723
E(X)= 4.3
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:
Probability:
a
1.00
a*1=a
Value of 5:
Probability:
5
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
P (Probability)
0.35
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.10
0.15
0.10
0.05
0.00
100
120
140
160
180
200
Value of X (weight in pounds)
Value of X
(Weight in pounds):
Probability:
100
0.10
120
0.15
140
0.30
160
0.25
180
0.20
E(X)=146 lb.
But wait!! The scale was off by 20 lb! Have to add 20 to all values...
Adding a Constant
P (Probability)
0.35
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.10
0.15
0.10
0.05
0.00
100
120
140
160
180
200
Value of X (weight in pounds)
Value of X
(Weight in pounds):
Probability:
100
0.10
120
0.15
140
0.30
160
0.25
180
0.20
160
0.30
180
0.25
200
0.20
E(X)=146 lb.
Value of X
(Weight in pounds):
Probability:
120
0.10
E(X)= 166 lb. =146+20
140
0.15
E(X+a) = E(X) + a
Adding a Constant
P (Probability)
(Probability)
P
0.35
0.35
0.30
0.30
0.30
0.30
0.25
0.25
0.25
0.25
0.10
0.10
0.25
0.20
0.20
0.20
0.20
0.15
0.15
0.30
0.15
0.15
0.10
0.10
0.20
0.15
0.10
0.05
0.05
0.00
0.00
100
100
120
120
140
140
160
160
180
180
200
200
Value
(weightinin
pounds)
Value of
of X (weght
pounds)
Note: the whole
distribution shifts to the right, but it doesn’t
Value of X
change(Weight
shape!in The
variance
(spread) 120
stays the140same. 160
pounds):
100
180
Probability:
0.10
0.15
0.30
0.25
0.20
180
0.25
200
0.20
Var(X±a)
= Var(X)
E(X)=146
lb.
Value of X
(Weight in pounds):
Probability:
120
0.10
E(X)= 166 lb. =146+20
140
0.15
160
0.30
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
P (Probability)
0.35
0.30
0.30
0.25
0.25
0.25
0.20
0.15
0.10
0.10
0.10
0.05
0.00
1.5
1.6
1.7
1.8
1.9
Value of X (height in meters)
Value of X
(height in meters)
Probability:
1.5
0.10
1.6
0.25
1.7
0.30
1.8
0.25
1.9
0.10
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
P (Probability)
0.35
0.30
0.30
0.25
0.25
0.25
0.20
0.15
0.10
0.10
0.10
0.05
0.00
1.5
1.6
1.7
1.8
1.9
Value of X (height in meters)
Value of X
(height in meters)
Probability:
1.5
0.10
1.6
0.25
1.7
0.30
1.8
0.25
1.9
0.10
5.58
0.30
5.90
0.25
6.23
0.10
E(X)=1.7 meters
Value of X
(height in feet):
Probability:
4.92
0.10
E(X)= 5.58 ft =3.28*1.7
5.25
0.25
E(aX) = a*E(X)
Multiplying by a Constant
P (Probability)
0.35
0.30
0.30
0.25
0.25
0.25
0.20
0.15
0.10
[Draw
new distribution on chalkboard.]0.10
0.10
0.05
Note: 0.00
the whole distribution shifts to the right, AND it gets
1.5
1.6
1.7
1.8
1.9
more spread out!
The variance
has
increased!
Value of X (height in meters)
Value of X
(height in meters)
Probability:
2*Var(X)
Var(aX)
=
a
1.5
1.6
1.7
0.10
0.25
0.30
1.8
0.25
1.9
0.10
5.58
0.30
5.90
0.25
6.23
0.10
E(X)=1.7 meters
Value of X
(height in feet):
Probability:
4.92
0.10
E(X)= 5.58 ft =3.28*1.7
5.25
0.25
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
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.
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)
 E(XY)=E(X)*E(Y)
Var(X) = E(X2) - (E(X))2
E(X2) = Var(X) + (E(X))2
Expected Value & Variance:
ALEKS Example
Rando m varia bles (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
Cumu lative distribution function
Exp ectation and variance of a random variable
Rules for expectation and variance of random variables
Two random va riables (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)
QuickTime™ and a
decompressor
are needed to see this picture.
algebra!
Var(aX)
Var(X±a)
Var(X) = E(X2) - (E(X))2
E(X2) = Var(X) + (E(X))2
QuickTime™ and a
decompressor
are needed to see this picture.
QuickTime™ and a
decompressor
are needed to see this picture.
QuickTime™ and a
decompressor
are needed to see this picture.