Random Variables
2.1 Discrete Random Variables2.2 The Expectation of a Random Variable2.3 The Variance of a Random Variable2.4 Jointly Distributed Random Variables2.5 Combinations and Functions of Random Variables
Random Variables
• Random Variable (RV): A numeric outcome that results from an experiment
• For each element of an experiment’s sample space, the random variable can take on exactly one value
• Discrete Random Variable: An RV that can take on only a finite or countably infinite set of outcomes
• Continuous Random Variable: An RV that can take on any value along a continuum (but may be reported “discretely”
• Random Variables are denoted by upper case letters (Y)
• Individual outcomes for RV are denoted by lower case letters (y)
Probability Distributions
• Probability Distribution: Table, Graph, or Formula that describes values a random variable can take on, and its corresponding probability (discrete RV) or density (continuous RV)
• Discrete Probability Distribution: Assigns probabilities (masses) to the individual outcomes
• Continuous Probability Distribution: Assigns density at individual points, probability of ranges can be obtained by integrating density function
• Discrete Probabilities denoted by: p(y) = P(Y=y)• Continuous Densities denoted by: f(y)• Cumulative Distribution Function: F(y) = P(Y≤y)
Discrete Probability Distributions
yyF
FF
ypbYPbF
yYPyF
yp
yyp
yYPyp
b
y
y
in increasinglly monotonica is )(
1)(0)(
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Example – Rolling 2 Dice (Red/Green)
Red\Green 1 2 3 4 5 6
1 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 94 5 6 7 8 9 105 6 7 8 9 10 116 7 8 9 10 11 12
Y = Sum of the up faces of the two die. Table gives value of y for all elements in S
Rolling 2 Dice – Probability Mass Function & CDF
y p(y) F(y)
2 1/36 1/36
3 2/36 3/36
4 3/36 6/36
5 4/36 10/36
6 5/36 15/36
7 6/36 21/36
8 5/36 26/36
9 4/36 30/36
10 3/36 33/36
11 2/36 35/36
12 1/36 36/36
y
t
tpyF
yyp
2
)()(
inresult can die 2 waysof #
tosumcan die 2 waysof #)(
Rolling 2 Dice – Probability Mass Function
Dice Rolling Probability Function
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
2 3 4 5 6 7 8 9 10 11 12
y
p(y
)
Rolling 2 Dice – Cumulative Distribution Function
Dice Rolling - CDF
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7 8 9 10 11 12 13
y
F(y
)
Expected Values of Discrete RV’s
• Mean (aka Expected Value) – Long-Run average value an RV (or function of RV) will take on
• Variance – Average squared deviation between a realization of an RV (or function of RV) and its mean
• Standard Deviation – Positive Square Root of Variance (in same units as the data)
• Notation:– Mean: E(Y) = – Variance: V(Y) = 2
– Standard Deviation:
Expected Values of Discrete RV’s
2
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all
22
all
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YEYE
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Expected Values of Linear Functions of Discrete RV’s
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Example – Rolling 2 Dice
y p(y) yp(y) y2p(y)
2 1/36 2/36 4/36
3 2/36 6/36 18/36
4 3/36 12/36 48/36
5 4/36 20/36 100/36
6 5/36 30/36 180/36
7 6/36 42/36 294/36
8 5/36 40/36 320/36
9 4/36 36/36 324/36
10 3/36 30/36 300/36
11 2/36 22/36 242/36
12 1/36 12/36 144/36
Sum 36/36=1.00
252/36=7.00
1974/36=54.833
4152.28333.5
8333.5)0.7(8333.54
)(
0.7)()(
2
212
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12
2
y
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ypyYE
yypYE
2.1 Discrete Random VariableDefinition of a Random Variable (1/2)
• Random variable – A numerical value to each outcome of a particular
experiment
S
0 21 3-1-2-3
R
Definition of a Random Variable (2/2)
• Example 1 : Machine Breakdowns– Sample space : – Each of these failures may be associated with a repair
cost– State space : – Cost is a random variable : 50, 200, and 350
{ , , }S electrical mechanical misuse
{50,200,350}
Probability Mass Function (1/2)
• Probability Mass Function (p.m.f.)– A set of probability value assigned to each of the
values taken by the discrete random variable– and – Probability :
ip
ix
0 1ip 1iip
( )i iP X x p
Probability Mass Function (1/2)
• Example 1 : Machine Breakdowns– P (cost=50)=0.3, P (cost=200)=0.2,
P (cost=350)=0.5– 0.3 + 0.2 + 0.5 =1 50 200 350
0.3 0.2 0.5
ix
ip( )f x
0.5
0.3
50 200 350 Cost($)
0.2
Cumulative Distribution Function (1/2)
• Cumulative Distribution Function– Function : – Abbreviation : c.d.f
( ) ( )F x P X x :
( ) ( )y y x
F x P X y
( )F x1.0
0.5
0.3
50 200 3500 ($cost)x
Cumulative Distribution Function (2/2)
• Example 1 : Machine Breakdowns
50 ( ) (cost ) 0
50 200 ( ) (cost ) 0.3
200 350 ( ) (cost ) 0.3 0.2 0.5
350 ( ) (cost ) 0.3 0.2 0.5 1.0
x F x P x
x F x P x
x F x P x
x F x P x
Cumulative Distribution Function (1/3)
• Cumulative Distribution Function
( ) ( ) ( )
( )( )
xF x P X x f y dy
dF xf x
dx
( ) ( ) ( )
( ) ( )
( ) ( )
P a X b P X b P X a
F b F a
P a X b P a X b
2.3 The Expectation of a Random VariableExpectations of Discrete Random Variables (1/2)
• Expectation of a discrete random variable with p.m.f
• The expected value of a random variable is also called the mean of the random variable
( )i iP X x p
( ) i ii
E X p x
state space( ) ( )E X xf x dx
Expectations of Discrete Random Variables (2/2)
• Example 1 (discrete random variable)– The expected repair cost is
(cost) ($50 0.3) ($200 0.2) ($350 0.5) $230E
2.4 The variance of a Random VariableDefinition and Interpretation of Variance (1/2)
• Variance( )– A positive quantity that measures the spread of the
distribution of the random variable about its mean value– Larger values of the variance indicate that the
distribution is more spread out
– Definition:
• Standard Deviation– The positive square root of the variance– Denoted by
2
2 2
Var( ) (( ( )) )
( ) ( ( ))
X E X E X
E X E X
2
Definition and Interpretation of Variance (2/2)
2
2 2
2 2
2 2
Var( ) (( ( )) )
( 2 ( ) ( ( )) )
( ) 2 ( ) ( ) ( ( ))
( ) ( ( ))
X E X E X
E X XE X E X
E X E X E X E X
E X E X
Two distribution with identical mean values but different variances
( )f x
x
Examples of Variance Calculations (1/1)
• Example 1
2 2
2 2 2
2
Var( ) (( ( )) ) ( ( ))
0.3(50 230) 0.2(200 230) 0.5(350 230)
17,100
i ii
X E X E X p x E X
17,100 130.77
2.5 Jointly Distributed Random VariablesJointly Distributed Random Variables (1/4)
• Joint Probability Distributions– Discrete
( , ) 0
satisfying 1
i j ij
iji j
P X x Y y p
p
Jointly Distributed Random Variables (2/4)
• Joint Cumulative Distribution Function– Discrete
( , ) ( , )i jF x y P X x Y y
: :
( , )i j
iji x x j y y
F x y p
Jointly Distributed Random Variables (3/4)
• Example 19 : Air Conditioner Maintenance– A company that services air conditioner units in
residences and office blocks is interested in how to schedule its technicians in the most efficient manner
– The random variable X, taking the values 1,2,3 and 4, is the service time in hours
– The random variable Y, taking the values 1,2 and 3, is the number of air conditioner units
Jointly Distributed Random Variables (4/4)
• Joint p.m.f
• Joint cumulative distribution function
Y=number of units
X=service time
1 2 3 4
1 0.12 0.08 0.07 0.05
2 0.08 0.15 0.21 0.13
3 0.01 0.01 0.02 0.07
0.12 0.18
0.07 1.00
iji j
p
11 12 21 22(2,2)
0.12 0.18 0.08 0.15
0.43
F p p p p
Conditional Probability Distributions (1/2)
• Conditional probability distributions– The probabilistic properties of the random variable X
under the knowledge provided by the value of Y– Discrete
|
( , )( | )
( )ij
i jj
pP X i Y jp P X i Y j
P Y j p
Conditional Probability Distributions (2/2)
• Example 19– Marginal probability distribution of Y
– Conditional distribution of X
3( 3) 0.01 0.01 0.02 0.07 0.11P Y p
131| 3
3
0.01( 1| 3) 0.091
0.11Y
pp P X Y
p
Covariance
• Covariance
– May take any positive or negative numbers.– Independent random variables have a covariance of zero– What if the covariance is zero?
Cov( , ) (( ( ))( ( )))
( ) ( ) ( )
X Y E X E X Y E Y
E XY E X E Y
Cov( , ) (( ( ))( ( )))
( ( ) ( ) ( ) ( ))
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
X Y E X E X Y E Y
E XY XE Y E X Y E X E Y
E XY E X E Y E X E Y E X E Y
E XY E X E Y
Independence and Covariance
• Example 19 (Air conditioner maintenance)
( ) 2.59, ( ) 1.79E X E Y 4 3
1 1
( )
(1 1 0.12) (1 2 0.08)
(4 3 0.07) 4.86
iji j
E XY ijp
Cov( , ) ( ) ( ) ( )
4.86 (2.59 1.79) 0.224
X Y E XY E X E Y
