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Introduction to Probability
Chandrashekar L
Department of Computer Science and AutomationIndian Institute of [email protected]
Coins, Dice and Cards
What is the Probability of obtaining
� Atleast 1 Tail in 2 Tosses of a coin.
Coins, Dice and Cards
What is the Probability of obtaining
� Atleast 1 Tail in 2 Tosses of a coin.
34
(1)
� An odd number in a single roll of a die.
Coins, Dice and Cards
What is the Probability of obtaining
� Atleast 1 Tail in 2 Tosses of a coin.
34
(1)
� An odd number in a single roll of a die.
36
=12
(2)
� A ’Spade’ in a pack of shuffled cards.
Coins, Dice and Cards
What is the Probability of obtaining
� Atleast 1 Tail in 2 Tosses of a coin.
34
(1)
� An odd number in a single roll of a die.
36
=12
(2)
� A ’Spade’ in a pack of shuffled cards.
1352
=14
(3)
One step at a time
2 Tosses of a coin
� What all can come??
One step at a time
2 Tosses of a coin
� What all can come??HH;HT;TH;TT
One step at a time
2 Tosses of a coin
� What all can come??HH;HT;TH;TT
� What is that we are interested in??
One step at a time
2 Tosses of a coin
� What all can come??HH;HT;TH;TT
� What is that we are interested in??HT;TH;TT
One step at a time
2 Tosses of a coin
� What all can come??HH;HT;TH;TT
� What is that we are interested in??HT;TH;TT
� Probability of Event of interest
One step at a time
2 Tosses of a coin
� What all can come??HH;HT;TH;TT
� What is that we are interested in??HT;TH;TT
� Probability of Event of interest
number of favourable outcomesnumber of all possible outcomes
(4)
Some Terminology
� What all can come??
Some Terminology
� What all can come??
Sample Space S=ffHHg,fHTg,fTHg,fTTgg
Some Terminology
� What all can come??
Sample Space S=ffHHg,fHTg,fTHg,fTTgg
� What is that we may be interested in??
Some Terminology
� What all can come??
Sample Space S=ffHHg,fHTg,fTHg,fTTgg
� What is that we may be interested in??
Set of Events E
Some Terminology
� What all can come??
Sample Space S=ffHHg,fHTg,fTHg,fTTgg
� What is that we may be interested in??
Set of Events E
E =
n�fHHg
,�fHTg
,�fTHg
,�fTTg
,�fHHg,fHTg
,
�fHHg,fTHg
,�fHHg,fTTg
,�fHTg,fTHg
,�fHTg,fTTg
,
�fTHg,fTTg
,�fHHg,fHTg,fTHg
,�fHHg,fHTg,fTTg
,
�fHHg,fTHg,fTTg
,�fHTg,fTHg,fTTg
,
�fHHg,fHTg,fTHg,fTTg
,f�g
o
Some Terminology
� What all can come??
Sample Space S=ffHHg,fHTg,fTHg,fTTgg
� What is that we may be interested in??
Set of Events E
E =
n�fHHg
,�fHTg
,�fTHg
,�fTTg
,�fHHg,fHTg
,
�fHHg,fTHg
,�fHHg,fTTg
,�fHTg,fTHg
,�fHTg,fTTg
,
�fTHg,fTTg
,�fHHg,fHTg,fTHg
,�fHHg,fHTg,fTTg
,
�fHHg,fTHg,fTTg
,�fHTg,fTHg,fTTg
,
�fHHg,fHTg,fTHg,fTTg
,f�g
o
� Probability of Event of interest
Some Terminology
� What all can come??
Sample Space S=ffHHg,fHTg,fTHg,fTTgg
� What is that we may be interested in??
Set of Events E
E =
n�fHHg
,�fHTg
,�fTHg
,�fTTg
,�fHHg,fHTg
,
�fHHg,fTHg
,�fHHg,fTTg
,�fHTg,fTHg
,�fHTg,fTTg
,
�fTHg,fTTg
,�fHHg,fHTg,fTHg
,�fHHg,fHTg,fTTg
,
�fHHg,fTHg,fTTg
,�fHTg,fTHg,fTTg
,
�fHHg,fHTg,fTHg,fTTg
,f�g
o
� Probability of Event of interest
The probability assignment P : E ! [0;1℄
Abstraction
How to put an elephant inside a refrigrator?
Abstraction
How to put an elephant inside a refrigrator?
� Open the refrigrator.
Abstraction
How to put an elephant inside a refrigrator?
� Open the refrigrator.
� Put the Elephant inside.
Abstraction
How to put an elephant inside a refrigrator?
� Open the refrigrator.
� Put the Elephant inside.
� Close the refrigrator.
Abstraction
How to put an elephant inside a refrigrator?
� Open the refrigrator.
� Put the Elephant inside.
� Close the refrigrator.
How to compute probabilities
� Identify S.
Abstraction
How to put an elephant inside a refrigrator?
� Open the refrigrator.
� Put the Elephant inside.
� Close the refrigrator.
How to compute probabilities
� Identify S.
� Identify E.
Abstraction
How to put an elephant inside a refrigrator?
� Open the refrigrator.
� Put the Elephant inside.
� Close the refrigrator.
How to compute probabilities
� Identify S.
� Identify E.
� Do the assignment P.
Abstraction
How to put an elephant inside a refrigrator?
� Open the refrigrator.
� Put the Elephant inside.
� Close the refrigrator.
How to compute probabilities
� Identify S.
� Identify E.
� Do the assignment P.
Abstraction and Axioms
A probability space is a 3-tuple (S,E,P) with the followingconditions
Abstraction and Axioms
A probability space is a 3-tuple (S,E,P) with the followingconditions
1 0 � P(E1) � 1, for any event E1 2 E.
Abstraction and Axioms
A probability space is a 3-tuple (S,E,P) with the followingconditions
1 0 � P(E1) � 1, for any event E1 2 E.
2 P(S) = 1.
Abstraction and Axioms
A probability space is a 3-tuple (S,E,P) with the followingconditions
1 0 � P(E1) � 1, for any event E1 2 E.
2 P(S) = 1.
3 For any sequence E1;E2; : : : of disjoint sets (mutuallyexclusive) we have
P([iEi) =
X
i
P(Ei): (5)
Abstraction and Axioms
A probability space is a 3-tuple (S,E,P) with the followingconditions
1 0 � P(E1) � 1, for any event E1 2 E.
2 P(S) = 1.
3 For any sequence E1;E2; : : : of disjoint sets (mutuallyexclusive) we have
P([iEi) =
X
i
P(Ei): (5)
4 The set E satisfies the following conditions1 E1 2 E ) Ec
1 2 E.
Abstraction and Axioms
A probability space is a 3-tuple (S,E,P) with the followingconditions
1 0 � P(E1) � 1, for any event E1 2 E.
2 P(S) = 1.
3 For any sequence E1;E2; : : : of disjoint sets (mutuallyexclusive) we have
P([iEi) =
X
i
P(Ei): (5)
4 The set E satisfies the following conditions1 E1 2 E ) Ec
1 2 E.2 E1; E2 2 E ) (E1 \ E2) 2 E.
Abstraction and Axioms
A probability space is a 3-tuple (S,E,P) with the followingconditions
1 0 � P(E1) � 1, for any event E1 2 E.
2 P(S) = 1.
3 For any sequence E1;E2; : : : of disjoint sets (mutuallyexclusive) we have
P([iEi) =
X
i
P(Ei): (5)
4 The set E satisfies the following conditions1 E1 2 E ) Ec
1 2 E.2 E1; E2 2 E ) (E1 \ E2) 2 E.3 For any sequence E1; E2; : : : 2 E ) [
iEi 2 E.
Abstraction and Axioms
A probability space is a 3-tuple (S,E,P) with the followingconditions
1 0 � P(E1) � 1, for any event E1 2 E.
2 P(S) = 1.
3 For any sequence E1;E2; : : : of disjoint sets (mutuallyexclusive) we have
P([iEi) =
X
i
P(Ei): (5)
4 The set E satisfies the following conditions1 E1 2 E ) Ec
1 2 E.2 E1; E2 2 E ) (E1 \ E2) 2 E.3 For any sequence E1; E2; : : : 2 E ) [
iEi 2 E.
Certain Properties
� If E1;E2 2 E, with E1 � E2, then P(E1) � P(E2).
Certain Properties
� If E1;E2 2 E, with E1 � E2, then P(E1) � P(E2).
� P(E1 [ E2) = P(E1) + P(E2)� P(E1 \ E2).
Certain Properties
� If E1;E2 2 E, with E1 � E2, then P(E1) � P(E2).
� P(E1 [ E2) = P(E1) + P(E2)� P(E1 \ E2).
� P(S) = 1.
Certain Properties
� If E1;E2 2 E, with E1 � E2, then P(E1) � P(E2).
� P(E1 [ E2) = P(E1) + P(E2)� P(E1 \ E2).
� P(S) = 1.
� P(�) = 0.
Why Abstract
E1
E2
E3
S
Figure: Abstract Probability Space
� World is not as simple as Coins, Dice and Cards.
Why Abstract
E1
E2
E3
S
Figure: Abstract Probability Space
� World is not as simple as Coins, Dice and Cards.� Any abstraction is just as useful as
(a + b)2 = a2 + b2 + 2ab.
Conditional Probability
E1
E2
E3
E4
E5
S
A
Here we are interested in probabilities of events given an eventA has occured.
Conditional Probability
E1
E2
E3
E4
E5
S
A
Here we are interested in probabilities of events given an eventA has occured.
P(E1jA) =P(E1 \ A)
P(A)(6)
Conditional Probability
E1
E2
E3
E4
E5
S
A
Here we are interested in probabilities of events given an eventA has occured.
P(E1jA) =P(E1 \ A)
P(A)(6)
Independent Events
Events E1 and E2 are independent when
P(E1 \ E2) = P(E1)� P(E2): (7)
Independent Events
Events E1 and E2 are independent when
P(E1 \ E2) = P(E1)� P(E2): (7)
What is the probability of obtaining atleast one ‘Tail’ when acoin is Tossed twice. The probability of obtaining ‘Tail’ in asingle Toss is p.
Independent Events
Events E1 and E2 are independent when
P(E1 \ E2) = P(E1)� P(E2): (7)
What is the probability of obtaining atleast one ‘Tail’ when acoin is Tossed twice. The probability of obtaining ‘Tail’ in asingle Toss is p.
P(ffTTg,fHTg,fTHgg) = p � p + (1� p)� p + p � (1� p)(8)
Independent Events
Events E1 and E2 are independent when
P(E1 \ E2) = P(E1)� P(E2): (7)
What is the probability of obtaining atleast one ‘Tail’ when acoin is Tossed twice. The probability of obtaining ‘Tail’ in asingle Toss is p.
P(ffTTg,fHTg,fTHgg) = p � p + (1� p)� p + p � (1� p)(8)
Examples of independent events� Getting ‘Tail’ in the first toss is independent of getting
‘Head’ in the second toss.� When I toss a coin and roll a die simultaneously the
outcomes are independent of each other.� How many glasses of water I drink is independent of
whether it will rain Tomorrow.The above examples are Correct but Useless
Bayes’ Formula
E1;E2; : : : ;En be disjoint sets such that [iEi = S, let A be any
event
P(A) = P(AjE1)� P(E1) + P(AjE2)� P(E2) + : : :+ P(AjEn)� P(En)
(9)
Bayes’ Formula
E1;E2; : : : ;En be disjoint sets such that [iEi = S, let A be any
event
P(A) = P(AjE1)� P(E1) + P(AjE2)� P(E2) + : : :+ P(AjEn)� P(En)
(9)
This is similar to the cut-off mark calulations.
E1
E2
E3
E4
E5
S
A
Bayes’ Formula Contd
This leads to the following relation
P(AjB) =P(A \ B)
P(B)
=P(BjA)� P(A)
P(BjA)� P(A) + P(BjAc)� P(Ac): (10)
Independent Events Contd
S = [0;1℄;E1 = [0;12℄;E2 = [0;
14℄ [ [
12;34℄;E3 = [
14;34℄: (11)
E3
E2
E1
S
Independent Events Contd
Random Variable
A random variable is not
� Random.
Random Variable
A random variable is not
� Random.
� A Variable.
What is it?
Random Variable
A random variable is not
� Random.
� A Variable.
What is it?It is a function X : S ! R.
S
x1
E2x2
E1
X
Coin, Dice and Cards
Rs.1 on getting an odd number and 2 on getting a even numberon roll of a die. We can define X as follows
Coin, Dice and Cards
Rs.1 on getting an odd number and 2 on getting a even numberon roll of a die. We can define X as follows
S =f1;2;3;4;5;6g
X (1) =1
X (2) =2
X (3) =1
X (4) =2
X (5) =1
X (6) =2
(12)
Coin, Dice and Cards
One can define another R.V. Y where Rs.x=2 when x shows up.
Coin, Dice and Cards
One can define another R.V. Y where Rs.x=2 when x shows up.
S =f1;2;3;4;5;6g
Y (1) =12
Y (2) =1
Y (3) =32
Y (4) =2
Y (5) =52
Y (6) =3
(13)
P(X = 2), P(X = 1;Y = 1), P(X = 1;Y = �1).
Probability Mass Function
Associated with a R.V. X we are interested in its probabilitymass function (pmf)
Probability Mass Function
Associated with a R.V. X we are interested in its probabilitymass function (pmf)fX (x) = P(X = x)
Probability Mass Function
Associated with a R.V. X we are interested in its probabilitymass function (pmf)fX (x) = P(X = x)Let us compute the pmfs for X and Y .
fX (1) = P(X = 1) =12; fX (2) = P(X = 2) =
12
(14)
fY (1) = P(Y = 1) =12; fY (2) = P(Y = 2) =
12
fY (3) = P(Y = 3) =12; fY (4) = P(Y = 4) = 0
fY (5) = P(Y = 5) =0; fY (6) = P(Y = 6) = 0
fY (12) = P(Y =
12) =
12; fY (
32) = P(Y =
32) =
12
fY (52) = P(Y =
52) =
12
(15)
f (x) can also be thought of as the frequency of occurence of x .
Cumulative Distribution Function
Sometimes we are interested in the quantity F (x) = P(X � x).
Cumulative Distribution Function
Sometimes we are interested in the quantity F (x) = P(X � x).This is the cumulative distribution function
0 0:5 1 1:5 2 2:5
0
0:2
0:4
0:6
0:8
1
x
FX
(x
)
cdf of X
0 1 2 3
0
0:2
0:4
0:6
0:8
1
y
FY
(y)
cdf of Y
Useful Random Variables
Bernoulli Random Variable X is either Success-1 or Failure-0.Thus X takes values 1 or 0.
P(X = 0) = 1� p;
P(X = 1) = p;0 � p � 1:
Useful Random Variables
Bernoulli Random Variable X is either Success-1 or Failure-0.Thus X takes values 1 or 0.
P(X = 0) = 1� p;
P(X = 1) = p;0 � p � 1:
Binomial Random Variable X is the number Successes inn-independent trials. X takes values from 1 to n with
P(X = k) =�
nk
�
pk (1� p)n�k : (16)
Useful Random Variables
Bernoulli Random Variable X is either Success-1 or Failure-0.Thus X takes values 1 or 0.
P(X = 0) = 1� p;
P(X = 1) = p;0 � p � 1:
Binomial Random Variable X is the number Successes inn-independent trials. X takes values from 1 to n with
P(X = k) =�
nk
�
pk (1� p)n�k : (16)
Geometric Random Variable X is the number of trials neededto obtain Success. X takes values 1;2;3; : : : with
P(x = k) = (1� p)k�1p: (17)
Useful Random Variables
Bernoulli Random Variable X is either Success-1 or Failure-0.Thus X takes values 1 or 0.
P(X = 0) = 1� p;
P(X = 1) = p;0 � p � 1:
Binomial Random Variable X is the number Successes inn-independent trials. X takes values from 1 to n with
P(X = k) =�
nk
�
pk (1� p)n�k : (16)
Geometric Random Variable X is the number of trials neededto obtain Success. X takes values 1;2;3; : : : with
P(x = k) = (1� p)k�1p: (17)
Poisson Random Variable X takes values 0;1;2;3; : : : with
P(X = k) = exp(��)(�)k
k !(18)
Useful to model number of arrivals in unit time.
Useful Random Variables
0 0:2 0:4 0:6 0:8 10:2
0:4
0:6
0:8
x
p(X
Bernoulli
0 2 4 6 8 10
0
0:1
0:2
0:3
xp(
X
Binomial
0 2 4 6 8 10
0
0:1
0:2
0:3
0:4
x
p(X
Poisson
2 4 6 8 10
0
0:2
0:4
0:6
0:8
x
p(X
Geometric
Continuous Random Variables
� Till now the Random Variables assumed only discretevalues (finite or countable).
Continuous Random Variables
� Till now the Random Variables assumed only discretevalues (finite or countable).
� Consider a Random Variable that takes a continuum ofvalues.
Continuous Random Variables
� Till now the Random Variables assumed only discretevalues (finite or countable).
� Consider a Random Variable that takes a continuum ofvalues.
� Then P(X = x) is 0 for all x and pmf does not make sense.
Continuous Random Variables
� Till now the Random Variables assumed only discretevalues (finite or countable).
� Consider a Random Variable that takes a continuum ofvalues.
� Then P(X = x) is 0 for all x and pmf does not make sense.
� However the cdf still makes sense.
Continuous Random Variables
� Till now the Random Variables assumed only discretevalues (finite or countable).
� Consider a Random Variable that takes a continuum ofvalues.
� Then P(X = x) is 0 for all x and pmf does not make sense.
� However the cdf still makes sense.
� Equivalent to the pmf we define the probability densityfunction (pdf).
Continuous Random Variables
� Till now the Random Variables assumed only discretevalues (finite or countable).
� Consider a Random Variable that takes a continuum ofvalues.
� Then P(X = x) is 0 for all x and pmf does not make sense.
� However the cdf still makes sense.
� Equivalent to the pmf we define the probability densityfunction (pdf).
� The pdf is derivative of the cdf.
Useful C.R.V
Uniform R.V X , where X takes values in (a;b).
Useful C.R.V
Uniform R.V X , where X takes values in (a;b).f (x) = 1
b�a ; x 2 (a;b) and fX (x) = 0; x =2 (a;b)
Useful C.R.V
Uniform R.V X , where X takes values in (a;b).f (x) = 1
b�a ; x 2 (a;b) and fX (x) = 0; x =2 (a;b)Gaussian R.V, where X takes values from �1 to +1.
Useful C.R.V
Uniform R.V X , where X takes values in (a;b).f (x) = 1
b�a ; x 2 (a;b) and fX (x) = 0; x =2 (a;b)Gaussian R.V, where X takes values from �1 to +1.
f (x) =1
p2��
exp( x��) 2
2�2 (19)
Exponential R.V, where X takes values in [0;1),
Useful C.R.V
Uniform R.V X , where X takes values in (a;b).f (x) = 1
b�a ; x 2 (a;b) and fX (x) = 0; x =2 (a;b)Gaussian R.V, where X takes values from �1 to +1.
f (x) =1
p2��
exp( x��) 2
2�2 (19)
Exponential R.V, where X takes values in [0;1),
f (x) = �exp��x8x � 0: (20)
Gamma R.V, where X takes values in [0;1),
Useful C.R.V
Uniform R.V X , where X takes values in (a;b).f (x) = 1
b�a ; x 2 (a;b) and fX (x) = 0; x =2 (a;b)Gaussian R.V, where X takes values from �1 to +1.
f (x) =1
p2��
exp( x��) 2
2�2 (19)
Exponential R.V, where X takes values in [0;1),
f (x) = �exp��x8x � 0: (20)
Gamma R.V, where X takes values in [0;1),
f (x) =�exp��x(�x)(n�1)
(n � 1)!: (21)
Useful Continuous Random Variables
0 0:2 0:4 0:6 0:8 1
0:9
1
1:1
1:2
x
f(x)
Uniform
�6 �4 �2 0 2 4 6
0
0:1
0:2
0:3
0:4
xf(
x)
Gaussian
�5 0 5 10 15 20
0
5 � 10�2
0:1
x
f(x)
Gamma
�5 0 5 10 15 20
0
0:2
0:4
0:6
0:8
x
f(X
)
Exponential
Mean and Variance
If X is a D.R.V. , the mean or Expectation of X is
E[X ℄ =X
x
xP(X = x): (22)
Mean and Variance
If X is a D.R.V. , the mean or Expectation of X is
E[X ℄ =X
x
xP(X = x): (22)
If X is a C.R.V. , the mean or Expectation of X is
E[X ℄ =
1Z
�1
xf (x)dx : (23)
Mean and Variance
If X is a D.R.V. , the mean or Expectation of X is
E[X ℄ =X
x
xP(X = x): (22)
If X is a C.R.V. , the mean or Expectation of X is
E[X ℄ =
1Z
�1
xf (x)dx : (23)
Properties of Mean
� E(g(X )) =P
xg(x)P(X = x) for D.R.V.,
E(g(X )) =1R
�1g(x)f (x)dx .
Mean and Variance
If X is a D.R.V. , the mean or Expectation of X is
E[X ℄ =X
x
xP(X = x): (22)
If X is a C.R.V. , the mean or Expectation of X is
E[X ℄ =
1Z
�1
xf (x)dx : (23)
Properties of Mean
� E(g(X )) =P
xg(x)P(X = x) for D.R.V.,
E(g(X )) =1R
�1g(x)f (x)dx .
� E(aX + bY ) = aE(X ) + bE(Y ).
Mean and Variance
If X is a D.R.V. , the mean or Expectation of X is
E[X ℄ =X
x
xP(X = x): (22)
If X is a C.R.V. , the mean or Expectation of X is
E[X ℄ =
1Z
�1
xf (x)dx : (23)
Properties of Mean
� E(g(X )) =P
xg(x)P(X = x) for D.R.V.,
E(g(X )) =1R
�1g(x)f (x)dx .
� E(aX + bY ) = aE(X ) + bE(Y ).
Variance
� The Variance of X is given by Var(X ) = E[(X � E(X ))2℄
Variance
� The Variance of X is given by Var(X ) = E[(X � E(X ))2℄
� Var(X ) is always positive.
� Verify that Var(X ) = E(X 2)� (E(X ))2.
Two Random Variables
S
x1
E2x2
E1
X Y
S
E3
E4y2
y1
X ; Y
S
x1; y1
E3
E4
E2x2; y2
E1
Two Random Variables
We have joint cdf, pmf, pdf defined as follows
� Joint cdf FXY (x ; y) = P(X � x ;Y � y).
Two Random Variables
We have joint cdf, pmf, pdf defined as follows
� Joint cdf FXY (x ; y) = P(X � x ;Y � y).
� Joint pmf meaning fXY (x ; y) = P(X = x ;Y = y).
Two Random Variables
We have joint cdf, pmf, pdf defined as follows
� Joint cdf FXY (x ; y) = P(X � x ;Y � y).
� Joint pmf meaning fXY (x ; y) = P(X = x ;Y = y).
� Joint pdf fXY (x ; y), where FXY (x ; y) =yR
�1
xR
�1fXY (x ; y)dxdy .
Two Random Variables
We have joint cdf, pmf, pdf defined as follows
� Joint cdf FXY (x ; y) = P(X � x ;Y � y).
� Joint pmf meaning fXY (x ; y) = P(X = x ;Y = y).
� Joint pdf fXY (x ; y), where FXY (x ; y) =yR
�1
xR
�1fXY (x ; y)dxdy .
We have the marginal cdfs, pdfs and pmfs as follows.
� FX (x) = P(X � x) =1R
�1fXY (x ; y)dy for a C.R.V.
Two Random Variables
We have joint cdf, pmf, pdf defined as follows
� Joint cdf FXY (x ; y) = P(X � x ;Y � y).
� Joint pmf meaning fXY (x ; y) = P(X = x ;Y = y).
� Joint pdf fXY (x ; y), where FXY (x ; y) =yR
�1
xR
�1fXY (x ; y)dxdy .
We have the marginal cdfs, pdfs and pmfs as follows.
� FX (x) = P(X � x) =1R
�1fXY (x ; y)dy for a C.R.V.
� fX (x) = P(X = x) =P
yP(X = x ;Y = y). for a D.R.V.
Two Random Variables
We have joint cdf, pmf, pdf defined as follows
� Joint cdf FXY (x ; y) = P(X � x ;Y � y).
� Joint pmf meaning fXY (x ; y) = P(X = x ;Y = y).
� Joint pdf fXY (x ; y), where FXY (x ; y) =yR
�1
xR
�1fXY (x ; y)dxdy .
We have the marginal cdfs, pdfs and pmfs as follows.
� FX (x) = P(X � x) =1R
�1fXY (x ; y)dy for a C.R.V.
� fX (x) = P(X = x) =P
yP(X = x ;Y = y). for a D.R.V.
Given the Marginals the joint distribution cannot be determinedin general.
Two Random Variables
We have joint cdf, pmf, pdf defined as follows
� Joint cdf FXY (x ; y) = P(X � x ;Y � y).
� Joint pmf meaning fXY (x ; y) = P(X = x ;Y = y).
� Joint pdf fXY (x ; y), where FXY (x ; y) =yR
�1
xR
�1fXY (x ; y)dxdy .
We have the marginal cdfs, pdfs and pmfs as follows.
� FX (x) = P(X � x) =1R
�1fXY (x ; y)dy for a C.R.V.
� fX (x) = P(X = x) =P
yP(X = x ;Y = y). for a D.R.V.
Given the Marginals the joint distribution cannot be determinedin general.We also can talk about the conditional fX jY=y (x), distribution ofX given that Y assumes a value y .
Independent Random Variables
� Joint cdf
FXY (x ; y) = P(X � x ;Y � y)
= P(X � x)P(Y � y)
= FX (x)FY (y)
Independent Random Variables
� Joint cdf
FXY (x ; y) = P(X � x ;Y � y)
= P(X � x)P(Y � y)
= FX (x)FY (y)
� The pdf (also the pmf) fXY (x ; y) = fX (x)fY (y).
In case of independent R.Vs
� We can determine the joints given the marginals.
� The conditional distribution is same as the marginal.
Identically Distributed R.Vs
Two R.Vs X and Y are said to be identically distributed if
Identically Distributed R.Vs
Two R.Vs X and Y are said to be identically distributed ifFX (x) = FY (x).Consider the following R.Vs defined on S = [0;1℄.
Identically Distributed R.Vs
Two R.Vs X and Y are said to be identically distributed ifFX (x) = FY (x).Consider the following R.Vs defined on S = [0;1℄.X (s) = 1; s 2 [0;0:5), X (s) = 0; s 2 [0:5;1℄.
Identically Distributed R.Vs
Two R.Vs X and Y are said to be identically distributed ifFX (x) = FY (x).Consider the following R.Vs defined on S = [0;1℄.X (s) = 1; s 2 [0;0:5), X (s) = 0; s 2 [0:5;1℄.Y = 1� X
P(X = 1) = 0:5;P(Y = 1) = 0:5
P(X = 0) = 0:5;P(Y = 0) = 0:5
(24)
Illustration with Dice
X -outcome of first toss, Y -outcome of second toss.
Illustration with Dice
X -outcome of first toss, Y -outcome of second toss.
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
YX
Illustration with Dice
Z -sum of the two outcomes.
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� Which R.Vs are independent?
� Which R.Vs are independent?
� Which R.Vs are identically distributed?
� Which R.Vs are independent?
� Which R.Vs are identically distributed?
� What is the marginal distribution of X , Y and Z?
� Which R.Vs are independent?
� Which R.Vs are identically distributed?
� What is the marginal distribution of X , Y and Z?
� Is fXZ (x ; z) = fX (x)fZ (z)?
� Which R.Vs are independent?
� Which R.Vs are identically distributed?
� What is the marginal distribution of X , Y and Z?
� Is fXZ (x ; z) = fX (x)fZ (z)?
� Is fXY (x ; y) = fX (x)fY (y)?
� Which R.Vs are independent?
� Which R.Vs are identically distributed?
� What is the marginal distribution of X , Y and Z?
� Is fXZ (x ; z) = fX (x)fZ (z)?
� Is fXY (x ; y) = fX (x)fY (y)?
� What is the conditional distributionfZ jX=3; fZ jX>3; fZ jX>3;Y<5; fZ jX=5;Y=5?
� Which R.Vs are independent?
� Which R.Vs are identically distributed?
� What is the marginal distribution of X , Y and Z?
� Is fXZ (x ; z) = fX (x)fZ (z)?
� Is fXY (x ; y) = fX (x)fY (y)?
� What is the conditional distributionfZ jX=3; fZ jX>3; fZ jX>3;Y<5; fZ jX=5;Y=5?
Conditional Mean and Covariance
The Conditional Mean denoted as E(X jY ) is a function of Y ,say h(Y ).
Conditional Mean and Covariance
The Conditional Mean denoted as E(X jY ) is a function of Y ,say h(Y ).
h(y) = E(X jy = y)
=X
xfX jY=y (x) for D.R.V
=
Z
xfX jY=y (x)dx for C.R.V
Conditional Mean and Covariance
The Conditional Mean denoted as E(X jY ) is a function of Y ,say h(Y ).
h(y) = E(X jy = y)
=X
xfX jY=y (x) for D.R.V
=
Z
xfX jY=y (x)dx for C.R.V
One can also check that E(h(Y )) = E(X ).
Conditional Mean and Covariance
The Conditional Mean denoted as E(X jY ) is a function of Y ,say h(Y ).
h(y) = E(X jy = y)
=X
xfX jY=y (x) for D.R.V
=
Z
xfX jY=y (x)dx for C.R.V
One can also check that E(h(Y )) = E(X ).The covariance between R.Vs X and Y is defined as
Conditional Mean and Covariance
The Conditional Mean denoted as E(X jY ) is a function of Y ,say h(Y ).
h(y) = E(X jy = y)
=X
xfX jY=y (x) for D.R.V
=
Z
xfX jY=y (x)dx for C.R.V
One can also check that E(h(Y )) = E(X ).The covariance between R.Vs X and Y is defined as
E�(X � E(X ))(Y � E(Y ))
�(25)
Approximation of a Random Variable X
� What is best constant that approximates X , i.e.,min
aE�(X � a)2
�
Approximation of a Random Variable X
� What is best constant that approximates X , i.e.,min
aE�(X � a)2
�
a = E(x):
� What is the best possible function of Y that approximatesX , i.e., min
gE�(X � g)2
�
Approximation of a Random Variable X
� What is best constant that approximates X , i.e.,min
aE�(X � a)2
�
a = E(x):
� What is the best possible function of Y that approximatesX , i.e., min
gE�(X � g)2
�
g = E(X jY ):
� What is the best possible linear function of Y thatapproximates X , i.e., min
a;bE�X � (bY + a)
�2
Approximation of a Random Variable X
� What is best constant that approximates X , i.e.,min
aE�(X � a)2
�
a = E(x):
� What is the best possible function of Y that approximatesX , i.e., min
gE�(X � g)2
�
g = E(X jY ):
� What is the best possible linear function of Y thatapproximates X , i.e., min
a;bE�X � (bY + a)
�2
b =CoVar(X ;Y )
Var(Y )a = E(X � bY )
Sum of Two Independent Random Variables
Let X and Y be independent R.Vs, Let Z = X + Y .
Sum of Two Independent Random Variables
Let X and Y be independent R.Vs, Let Z = X + Y .
fZ (z) =1R
�1fX (x)fY (z � x)dx
Sum of Two Independent Random Variables
Let X and Y be independent R.Vs, Let Z = X + Y .
fZ (z) =1R
�1fX (x)fY (z � x)dx
0 0:2 0:4 0:6 0:8 1
0:9
1
1:1
1:2
x
f(x)
X1
0 0:5 1 1:5
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f(x)
X1 + X2
0 0:5 1 1:5 2 2:5
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f(x)
X1 + X2 + X3
Sum of infinitely many random Variables
If X1;X2; : : : ;Xn : : : are independent identically distributedrandom variables
� limn!1
1n
nP
i=1Xi ! E(X1).
Sum of infinitely many random Variables
If X1;X2; : : : ;Xn : : : are independent identically distributedrandom variables
� limn!1
1n
nP
i=1Xi ! E(X1).
� If Y = limn!1
1pn
nP
i=1Xi . Then fY isN (0;1), i.e., Gaussian R.V
with mean 0 and variance 1.