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Probability Distributions and Probability Densities
Prob. Distributions & Densities - 1
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Probability Distributions and
Probability Densities
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1. Random Variable Definition: If S is a sample space with a
probability measure and X is a real-valued
function defined over the elements of S, then
X is called a random variable. [ X(s) = x]
Random variables are usually denoted by
capital letter. X, Y, Z, ...
Lower case of letters are usually for denoting
a element or a value of the random variable.
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Discrete Random Variable
Discrete Random Variable:
A random variable
assumes discrete values
by chance.
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Discrete Random Variable
Example: (Toss a balanced coin)
X = 1, if Head occurs, and
X = 0, if Tail occurs.
P(Head) = P(X = 1) = P(1) = .5
P(Tail) = P(X = 0) = P(0) = .5
Probability mass function:
f(x) = P(X = x) =.5 , if x = 0, 1,
and 0 elsewhere.
1/2
0 1
Probability
Bar Chart
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Why Random Variable?
• A simple mathematical notation to
describe an event. e.g.: X < 3, X = 0, ...
• Mathematical function can be used to
model the distribution through the use of
random variable. e.g.: Binomial, Poisson,
Normal, …
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Discrete Random Variable Example: (Toss a balanced coin 3 times)
X takes on number of heads occurs.
Outcomes Probability x
HHH 1/8 3
HHT 1/8 2
HTH 1/8 2
THH 1/8 2
HTT 1/8 1
THT 1/8 1
TTH 1/8 1
TTT 1/8 0
P(X=3) =1/8
P(X=2) =3/8
P(X=1) =3/8
P(X=0) =1/8
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 2
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Discrete Random Variable Example: (Toss a balanced coin 3 times)
X takes on number of heads occurs.
3,2,1,0,8
3
)()(
xx
xXPxf
Probability Distribution of X.
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2. Probability Distribution
(or Probability Mass Function)
If X is a discrete random variable, the function
f(x) = P(X=x) is called the probability distribution
[or probability mass function (p.m.f.) ] of X, and
this function satisfies the following properties:
a) f(x) > 0, for each value of in the space S of X,
b) SxS f(x) = 1.
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Probability Distribution
Example: Is f(x) a proper probability
distribution of the random variable X?
3. 1, 1, for ,3
)( xx
xf
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Probability Distribution
Example: Is f(x) a proper probability
distribution of the random variable X?
3. 2, 1, for ,3
)( xx
xf
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Probability Distribution
Example: If f(x) is a probability distribution
of a discrete random variable X, find the
value of c if
3. 2, 1, for ,
3)( xx
cxf
1)3()2()1()(3
1
fffxfi
Sol:
12
)3
3
3
2
3
1(3
32
31
3
c
cccc
2
1 c
Property 2
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Probability Distribution
Example: Two balls are to be selected from
a box containing 5 blue balls and 3 red balls
without replacement. Find the probability
distribution for the probability of getting x
blue balls.
Sol:
2
8
1
3
1
5
)1()1( XPf
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 3
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Probability Distribution
2. 1, 0, for ,
2
8
2
35
)(
xxx
xf
x f(x) = P(X = x)
2 20/56
1 30/56
0 15/56
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Is it a profitable insurance
premium?
Distribution: f (-100) = .1
f (-1) = .25,
f (11) = .65,
-100 -1 11
.75
.50
.25
Probability line chart
X=x f (x)
100 .1
1 .25
11 ?
100 ,1.
1 ,25.
11 ,65.
)(
xif
xif
xif
xf
Probability Distribution
.65
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Cumulative Distribution
Function
The cumulative distribution function (c.d.f.
or distribution function, d.f.) of a discrete
random variable is defined as
xt
xtfxXPxF .for ),()()(
• F( ) = 0, F() = 1
• If a < b, F(a) ≤ F(b) for any real numbers a, b.
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Distribution Function Example: (Toss a balanced coin 3 times)
X takes on number of heads occurs.
x f(x) F(x)
0 1/8 1/8
1 3/8 4/8
2 3/8 7/8
3 1/8 1
17
x
F(x)
1/2
1
0 1 2 3
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Theorem: If the range of a random variable X
consists of the values x1 < x2 < … < xn , then
f(x1) =F(x1), and f(xi) = F(xi) – F(xi-1), for i =
1, 2, 3, …
Example: (Toss a balanced coin 3 times)
X takes on number of heads occurs.
f(2) = F(2) – F(1) = 4/8 – 1/8 = 3/8
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 4
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Sample Space (Two Dice)
1. Outcome pairs:
S = {(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)}
2. Sum: S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Probability Distribution of the Sum
(Two Dice) Let X be a random variable takes on the sum of
the two dice.
P(X = 6) = ?
P(X = 8) = ?
P(X ≤ 8) = ?
20
21
Probability Distribution of the Sum
(Two Dice)
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Distribution Function of the Sum
(Two Dice)
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3.4 Continuous Random Variable &
Probability Density Function
Percent
A smooth curve that fit
the distribution
10 20 30 40 50 60 70 80 90 100 110
Test scores
Density
function, f (x)
f(x)
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Definition 3.4: A function with values f(x), defined
over the set of all real numbers, is called a probability
density function (p.d.f.) of the continuous random
variable X, if and only if
f(x)
x a b
P(a ≤ X ≤ b) =
for any real constants a and b with a ≤ b.
Notes:
f(c) is the probability density at c.
f(c) ≠ P(X = c) = 0 for any continuous random variable.
𝑓 𝑥 𝑑𝑥𝑏
𝑎
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 5
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Theorem 3.4: If X is a continuous random
variable and a and b are real constants with
a ≤ b, then
P(a ≤ X ≤ b) = P(a < X ≤ b) = P(a ≤ X <b) =
P(a < X < b)
Theorem 3.5: A function can serve a
probability density function of a continuous
random variable X if its values, f(x), satisfy
the conditions,
1. f(x) ≥ 0, for -∞ < x < ∞;
2. 𝑓 𝑥 𝑑𝑥∞
−∞ = 1.
26
Example: If X is a continuous random variable with
a p.d.f.
𝑓 𝑥 = 𝑘 ∙ 𝑒−3𝑥, for 𝑥 > 00, elsewhere
Find k.
Find P(0.5 < X < 1).
27
Definition 3.5: If X is a continuous random
variable and the value of its probability density
at t is f(t), then the function given by
𝐹 𝑥 = 𝑃 𝑋 ≤ 𝑥
= 𝑓 𝑡 𝑑𝑡, for −∞ < 𝑥 < ∞𝑥
−∞
is called the distribution function, or the
cumulative distribution of X, and
• F(−∞) = 0, F(∞) = 1,
• If a < b, F(a) ≤ F(b).
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Theorem 3.6: If f(x) and F(x) are values
of the probability density and the
distribution function of the random
variable X at x, then
P(a ≤ X ≤ b) = F(a) F(b)
for any real constants a and b with a ≤ b,
and
𝑓 𝑥 =𝑑𝐹(𝑥)
𝑑𝑥
where the derivative exists.
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Example: If X is a continuous random
variable with a p.d.f.
𝑓 𝑥 = 3𝑒−3𝑥, for 𝑥 > 00, elsewhere
Find the distribution function of X.
Use the d.f. above to find P(0.5 < X < 1).
30
Example: Find a probability density function for the
random variable whose distribution function is given
by
𝐹 𝑥 = 0, for 𝑥 ≤ 0𝑥, for 0<𝑥 < 11, for 𝑥 ≥ 1
and plot its graph.
Sol: The function is discontinuous at 0 and 1.
By Theorem 3.6
𝑓 𝑥 = 0, for 𝑥 ≤ 01, for 0<𝑥 < 10, for 𝑥 ≥ 1
We can let f(x) = 0 at 0 and 1, then we’ll have
𝑓 𝑥 = 1, for 0<𝑥 < 10, elsewhere.
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 6
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F(x)
1
¾
½
¼
0 0.5 1
x
A Mixed Distribution
What is its F(x)?
What is its f(x)?
32
F(x)
1
½
0 1 2 3
Example: X is a random variable with the following d.f.,
Find P(X < 2) =
P(X < 1/2) =
P(X = 1) =
P(X ≤ 3/2) =
P(X > 3/2) =
𝐹 𝑥 =
0, 𝑥 < 0𝑥/2, 0 ≤ 𝑥 < 13/4, 1 ≤ 𝑥 < 25/6, 2 ≤ 𝑥 < 31, 3 ≤ 𝑥.
Multivariate Distributions
33
3.5 Multivariate Distributions
Univariate (one random variable)
Bivariate (two random variables over a joint sample space)
Multivariate (two or more random variables over a joint sample space)
If X and Y are two discrete random variables, we
denote the probability of X takes on value x and Y
takes on value y as P(X = x, Y = y).
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Example: Two balls are selected from a box
containing 3 red balls, 2 blue balls, and 4 green balls.
If X and Y are, respectively, the numbers of red balls
and blue balls draw from the box, find the
probabilities associated with all possible pairs of
values of X and Y.
Sol: S = {(0,0), (0,1), (0,2), (1,0), (1,1), (2,0)}
6
1
2
9
2
4
0
2
0
3
)0,0(
P
35
Two balls are selected from a box containing 3 red
balls, 2 blue balls, and 4 green balls.
.20 ;2,1,0,,
2
9
2
423
),(
yxyxyxyx
yxP
2 1/36
y 1 2/9 1/6
0 1/6 1/3 1/12
0 1 2
x
x
Y
2
1
0
0 1 2 36
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 7
Definition 3.6. If X and Y are discrete random
variables, the function given by f(x, y) = P(X=x, Y=y)
for each pair of values (x, y) within the range of X
and Y is called the joint probability distribution of
X and Y.
37
Theorem 3.7: A bivariate function can serve the joint
probability distribution of a pair of discrete random
variables X and Y if and only if its values, f(x, y) ,
satisfy the conditions
• = 1, where the double summation
extends over all possible pairs (x, y) within its
domain.
• f(x, y) ≥ 0, for each pair of values (x, y) within
its domain; ∞ < x < ∞;
38
Example: Determine the value of k for which the
function given by f(x, y) = kxy, for x = 1, 2; and y =
1, 2, can serve as a joint probability distribution.
39
Example: In the example about drawing two
balls problem, find
• P(1 ≤ X ≤ 2, 0 ≤ Y ≤ 1) = ?
• P(X ≤ 1, 0 ≤ Y ≤ 1) = ?
2 1/36
y 1 2/9 1/6
0 1/6 1/3 1/12
0 1 2
x
1/6 + 1/3 + 1/12 = 7/12
2/9 + 1/6 + 1/6 + 1/3 = 7/9
40
Definition 3.7. If X and Y are discrete random
variables, the function given by
F(x, y) = P(X ≤ x, Y ≤ y) =
for ∞ < x < ∞, ∞ < y < ∞,
where f(s, t) is the value of the joint probability
distribution of X and Y at (s, t), is called the joint
distribution function, or the joint cumulative
distribution of X and Y.
xs yttsf ),,(
41
Example: In the last example about three colors
balls problem, find
F(1, 1) = ?
F(0.5, 1.2) = ?
F(1, 1) = ?
2 1/36
y 1 2/9 1/6
0 1/6 1/3 1/12
0 1 2
x
x
Y
2
1
0
0 1 2
42
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 8
Definition 3.8. A bivariate function f(x, y), define
over the xy-plane, is called a joint probability
density function of the continuous random variables
X and Y if and only if
for any region A in the xy-plane.
A
dxdyyxfAYXP ),(]),[(
43
Theorem 3.8: A bivariate function can serve as a joint
probability density function of a pair of continuous
random variables X and Y if its values, f(x, y) , satisfy
the conditions
• .
• f(x, y) ≥ 0, for ∞ < x < ∞; ∞ < y < ∞,
1),( dxdyyxf
44
Example: The following is the joint probability
density function of two continuous random variables
X and Y,
elsewhere. 0
20 ,1 0for )(5
3
),(
yxyxx
yxf
1. Find P(0 < X < Y, Y < 1).
45
Example: The following is the joint probability
density function of two continuous random variables
X and Y,
elsewhere. 0
20 ,1 0for )(5
3
),(
yxyxx
yxf
2. Find P(0 < X < 1/2, 1< Y < 2).
3. Find P(0 < X + Y < 1/2).
46
Example: The following is the joint probability
density function of two continuous random variables
X and Y,
elsewhere. 0
10 , 0for 1
),(
yyxy
yxf
Find P(X + Y > 1/2).
47
y y y Definition 3.9: If X and Y are continuous random
variables with joint probability density function f(x, y),
the function given by
for ∞< x < ∞; ∞< y < ∞, is called the joint
distribution function, or the joint cumulative
distribution of X and Y.
y x
dsdttsfyYxXPyxF ),(),(),(
),(),(2
yxFyx
yxf
48
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 9
Example: If the joint probability density of X and Y
is given by
find the joint distribution function of X and Y.
49
elsewhere. 0
10 ,1 0for ),(
yxyxyxf
I II
IV III
x
y
0 1
1
50
I II
IV III
x
y
0 1
1
F(x, y)
If x < 0 or y < 0, F(x, y) = 0
If 0 < x < 1 and 0 < y < 1,
F(x, y) = y x
dsdtts0 0
)(
dttssxy
|0
2
0 21 )(
(I)
|0
2
212
21 ][
y
xttx
dtxtxy
)( 2
0 21
2
212
21 xyyx )(
21 yxxy
51
I II
IV III
x
y
0 1
1
F(x, y)
If x > 1 and 0 < y < 1,
F(x, y) = y
dsdtts0
1
0)(
(II)
)1(21 yy
If 0 < x < 1 and y > 1,
F(x, y) = 1
0 0)(
x
dsdtts
(III)
)1(21 xx
If x > 1 and y > 1,
F(x, y) = 1
0
1
0)( dsdtts
(IV)
152
I II
IV III
x
y
0 1
1
F(x, y)
1,1for 1
1,10for )1(
10 ,1for )1(
10 ,10for )(
0,0for 0
),(
21
21
21
yx
yxxx
yxyy
yxyxxy
yx
yxF
53
Example: Find the joint probability density of the
two random variables X and Y whose joint
distribution function is given by
elsewhere 0
0 and 0for )1)(1(),(
yxeeyxF
yx
),(),(2
yxFyx
yxf
0 ,0for ,)( yxeee yxyx
and 0 elsewhere.
and also determine P(1 < X < 3, 1 < Y < 2).
Sol: For joint p.d.f.,
54
P(1 < X < 3, 1 < Y < 2) =
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 10
55
0 a b
d
c
(b, d)
(b, c)
(a, d)
(a, c)
x
y
P(a<X<b, c<Y<d)
= F(b,d) F(b,c) F(a,d) F(a,c)
56
Multivariate Discrete Distribution
The joint probability distribution of n discrete
random variables, X1, X2, X3, …, Xn, is
for each (x1, x2, x3, …, xn) in the range of the r.v.’s,
),...,,(),...,,( 221121 nnn xXxXxXPxxxf
),...,,(),...,,( 221121 nnn xXxXxXPxxxF
and the joint distribution function
for −∞ < x1< ∞, −∞ < x2 < ∞, …, −∞ < xn < ∞.
57
Example: If the joint probability density of three
random variables X ,Y, and Z is given by
elsewhere 0
1,2 and;3,2,1 ;2,1for 63
)(
),,(zyx
zyx
zyxf
find P(X = 2, Y + Z ≤ 3).
58
Multivariate Continuous Distribution
The joint probability density function of n
continuous random variables, X1, X2, X3, …, Xn, can
be define by a multivariate function ,
),...,,( 21 nxxxf
nn
x x x
nnn
dtdtdttttf
xXxXxXPxxxF
n
...),...,,(...
),...,,(),...,,(
2121
221121
2 1
and the joint distribution function is
for −∞ < x1< ∞, −∞ < x2 < ∞, …, −∞ < xn < ∞,
and
).,...,,(...
),...,,( 21
21
21 n
n
n
n xxxFxxx
xxxf
Example: If the joint probability density function of
X1, X2, and X3 is given by
find P(0 < X1 < 1/2, 1/2 < X2 < 1, X3 < 1).
59
elsewhere. 0
0 ,10 ,1 0for )(),,( 32121
321
3 xxxexxxxxf
x
Two balls are selected from a box containing 3 red
balls, 2 blue balls, and 4 green balls.
.20 ;2,1,0,,
2
9
2
423
),(
yxyxyxyx
yxP
2 1/36
y 1 2/9 1/6
0 1/6 1/3 1/12
0 1 2
x
x
Y
2
1
0
0 1 2 60
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 11
61
3.6 Marginal Distributions
x
Y
2
1
0
0 1 2
2 1/36
y 1 2/9 1/6
0 1/6 1/3 1/12
0 1 2
x
5/12 1/2 1/12
1/36
7/18
7/12
62
2 1/36
y 1 2/9 1/6
0 1/6 1/3 1/12
0 1 2
x
5/12 1/2 1/12
The Marginal Probability Distribution of X:
2
0
.2 ,1 ,0for ),,()(y
xyxfxg
.12/1)2( ,2/1)1( ,12/5)0( ggg
63
2 1/36
y 1 2/9 1/6
0 1/6 1/3 1/12
0 1 2
x
1/36
7/18
7/12
The Marginal Probability Distribution of Y:
2
0
.2 ,1 ,0for ),,()(x
yyxfyh
.36/1)2( ,18/7)1( ,12/7)0( hhh
Definition 3.11. If X and Y are discrete random
variables with joint probability distribution f(x, y),
the function given by
for each x in the space of X, is called the marginal
distribution of X. Correspondingly,
64
y
yxfxg ),()(
x
yxfyh ),()(
for each y in the space of Y, is called the marginal
distribution of Y.
Definition 3.11. If X and Y are continuous random
variables with joint probability density f(x, y), the
function given by
for x < is called the marginal distribution of
X. Correspondingly,
65
dyyxfxg
),()(
for y < is called the marginal distribution of
Y.
dxyxfyh
),()(
Example: Given the joint probability density
function of two continuous random variables X and Y,
elsewhere ,0
10 ,1 0for ),2(3
2
),(yxyx
yxf
find the marginal density of X and Y.
66
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 12
Example: Given the trivariate joint probability
density function random variables X1, X2, X3,
elsewhere, ,0
0 ,10 ,1 0for ,)(),,( 32121
321
3 xxxexxxxxf
x
find the marginal density of X1.
67
Example: Given the trivariate joint probability
density function random variables X1, X2, X3,
elsewhere ,0
0 ,10 ,1 0for ,)(),,( 32121
321
3 xxxexxxxxf
x
find the joint marginal density of X1 and X3.
68
Definition 3.11. If X and Y are discrete random
variables with joint probability distribution f(x, y),
the function given by
for each x in the space of X, is called the marginal
distribution of X. Correspondingly,
69
y
yxfxg ),()(
x
yxfyh ),()(
for each y in the space of Y, is called the marginal
distribution of Y.
Definition 3.11. If X and Y are continuous random
variables with joint probability density f(x, y), the
function given by
for x < is called the marginal probability
density of X. Correspondingly,
70
dyyxfxg
),()(
for y < is called the marginal probability
density of Y.
dxyxfyh
),()(
71
For more than two variables, if the discrete random
variables X1, X2, X3, …, Xn has joint probability
distribution f(x1, x2, x3, …, xn), then the marginal
probability distribution of X1 is
nx
n
x
xxxfxg ),...,,(...)( 211
2
for all x1 in the space of X1. The joint marginal
probability distribution of X1, X2, X3 is
nx
n
x
xxxfxxxm ),...,,(...),,( 21321
4
for all x1 , x2, x3, is in the space of X1, X2, X3.
72
For more than two variables, if the continuous
random variables X1, X2, X3, …, Xn has joint
probability density function f(x1, x2, x3, …, xn),
then the marginal density function of X1 is
nn dxdxxxxfxg ...),...,,(... )( 2211
for all x1 in the space of X1. The joint marginal
probability density of X1, X2, X3 is
nn dxdxxxxfxxxm ...),...,,(... ),,( 421321
for all x1 < , x2 < , …, xn < .
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 13
73
Joint Marginal Distribution Function
If F(x, y) is the joint distribution function of X
and Y, the marginal distribution function of X
is G(x) = F(x, ∞), for < x < .
If F(x1, x2, x3) is the joint distribution function of
X1, X2, X3, the joint marginal distribution
function of X1, X3, is M(x1, x3) = F(x1, ∞, x3), for
< x1 < , < x3 < .
Example: Given the joint distribution function of
two continuous random variables X and Y,
elsewhere, ,0
0 ,0for ),1)(1(),(
22
yxeeyxF
yx
find the marginal distribution of X.
74
elsewhere. ,0
0for ),1(),()(
2
xexFxG
x
x
dtdstsfxG ),()(
y x
dsdttsfyxF ),(),(
Example: Given the trivariate joint probability
density function random variables X1, X2, X3,
elsewhere, ,0
0 ,10 ,1 0for ,)(),,( 32121
321
3 xxxexxxxxf
x
find the marginal distribution of X1, G(x1).
75 76
3.7 Conditional Distribution
)(
)()|(
BP
BAPBAP
Discrete Case:
If A and B are events X = x and Y = y, then
)(
),(
)(
),()|(
yh
yxf
yYP
yYxXPyYxXP
Definition 3.12. If f(x, y) is the joint probability
distribution of the discrete random variables X and
Y, and h(y) is the marginal probability distribution
of Y at y, the function given by
is called the marginal conditional distribution of X
given Y = y. Correspondingly, if g(x) is the marginal
distribution of X at x, the function given by
77
0 ,)(
),()|( h(y)
yh
yxfyxf
is called the marginal conditional distribution of Y
given X = x.
0 ,)(
),()|( xg
xg
yxfxyw
Definition 3.13. If f(x, y) is the joint p.d.f. of the
continuous random variables X and Y, and h(y) is
the marginal probability density function of Y at y,
the function given by
is called the marginal conditional probability
density function of X given Y = y. Correspondingly,
if g(x) is the marginal p.d.f. of X at x, the function
given by
78
0 ,)(
),()|( h(y)
yh
yxfyxf
is called the marginal conditional probability
density function of Y given X = x.
0 ,)(
),()|( xg
xg
yxfxyw
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 14
79
2 1/36
y 1 2/9 1/6
0 1/6 1/3 1/12
0 1 2
x
5/12 1/2 1/12
Conditional probability distribution of X given Y = 1:
0)1|2(
7/318/7
6/1)1|1(
7/418/7
9/2)1|0(
f
f
f
1/36
7/18
7/12
80
2 1/36
y 1 2/9 1/6
0 1/6 1/3 1/12
0 1 2
x
5/12 1/2 1/12
Find P( X ≤ 1 | Y = 0 )
1/36
7/18
7/12
= P( X = 0 | Y = 0 ) + P( X = 1 | Y = 0 )
)0(
)0,1(
)0(
)0 ,0(
YP
YXP
YP
YXP
7
6
12/7
3/1
12/7
6/1
81
2 1/36
y 1 2/9 1/6
0 1/6 1/3 1/12
0 1 2
x
5/12 1/2 1/12
Find P( X ≤ 1 | Y ≤ 1 ) = ?
1/36
7/18
7/12
)1(
)1 ,1(
YP
YXP
12/718/7
3/16/16/19/2
Example: Given the joint probability density
function of two continuous random variables X and Y,
elsewhere ,0
,10 ,1 0for ),2(3
2
),(yxyx
yxf
find the marginal conditional density of X given
Y = y and use it to calculate P(X ≤ 1/2 | Y = 1/2).
82
elsewhere ,0
,10 ,1 0for ),2(3
2
),(yxyx
yxf
The marginal conditional density of X given Y = y
is
83
,10for ),41(3
1)( xyyh
elsewhere. 0)|( and
,10for ,41
42
)41(3
1
)2(3
2
)(
),()|(
yxf
xy
yx
y
yx
yh
yxfyxf
84
3
22
21
22)
2
1 (
xxxf
12
5
3
22
)2
1|()
2
1 |
2
1 (
21
21
0
0
dxx
dxxfYXP
Probability Distributions and Probability Densities
Prob. Distributions & Densities - 15
85
)2
1(
)2
1 ,
2
1 (
)2
1 |
2
1 (
YP
YXP
YXP
Can we do the following if both X and Y are
continuous random variables?
)2
1(
)2
1 ,
2
1 (
)2
1 |
2
1 (
YP
YXP
YXP
Can it be done in the following way? Example: Given the joint probability density
function of two continuous random variables X and Y,
elsewhere ,0
10 ,1 0for ,4),(
yxxyyxf
find the marginal density of X and Y and the
conditional p.d.f. of X given Y = y .
86
87
More than two variables:
.0)( ,)(
),,,()|,,(
.0),( ,),(
),,,(),|,(
.0),,( ,),,(
),,,(),,|(
),,,(,,,
1
1
43211432
43
43
43214321
432
432
43214321
43214321
xlxl
xxxxfxxxxr
xxmxxm
xxxxfxxxxq
xxxgxxxg
xxxxfxxxxp
xxxxfxxxx
Definition 3.14. If f(x1, x2, x3, …, xn) is the joint
probability distribution (density) of the discrete
(continuous) random variables X1, X2, X3, …, Xn,
and fi (xi) is the marginal probability distribution
(density) of Xi , for i = 1, 2, …, n, then the n
random variables are independent if and only if
f(x1, x2, x3, …, xn) = f1(x1) f2(x2) … fn(xn).
88
Example: Consider n independent flips of a
balanced coin, let Xi be the number of heads (0 or 1)
obtained in the i-th flip for i =1,2, …, n. Find the
joint probability distribution of these n random
variables.
89
Example: Given the independent random variables
X1, X2, and X3, with the probability density functions
elsewhere, ,0
0for ,)( 1
11
1 xexf
x
find the joint density of X1, X2, and X3, and also
find P(X1 + X2 ≤ 1, X3 > 1). 90
elsewhere, ,0
0for ,2)( 2
2
22
2 xexf
x
elsewhere, ,0
0for ,3)( 3
3
33
3 xexf
x