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Probability Distributions and Probability Densities
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]
Probability Distributions and
Probability Densities
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.
1
Discrete Random Variable
2
Discrete Random Variable
Example: (Toss a balanced coin)
Probability
Bar Chart
Discrete Random Variable:
X = 1, if Head occurs, and
X = 0, if Tail occurs.
A random variable
assumes discrete values
by chance.
P(Head) = P(X = 1) = P(1) = .5
P(Tail) = P(X = 0) = P(0) = .5
3
Why Random Variable?
1/2
0
1
Probability mass function:
f(x) = P(X = x) =.5 , if x = 0, 1,
and 0 elsewhere.
4
Discrete Random Variable
Example: (Toss a balanced coin 3 times)
• 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, …
X takes on number of heads occurs.
5
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
6
Prob. Distributions & Densities - 1
Probability Distributions and Probability Densities
2. Probability Distribution
(or Probability Mass Function)
Discrete Random Variable
Example: (Toss a balanced coin 3 times)
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:
X takes on number of heads occurs.
 3
 
x
f ( x)  P( X  x)    , x  0,1,2,3
8
a) f(x) > 0, for each value of in the space S of X,
b) SxS f(x) = 1.
Probability Distribution of X.
7
8
Probability Distribution
Probability Distribution
Example: Is f(x) a proper probability
distribution of the random variable X?
Example: Is f(x) a proper probability
distribution of the random variable X?
f ( x) 
x
, for x   1, 1, 3.
3
f ( x) 
x
, for x  1, 2, 3.
3
9
Probability Distribution
Probability Distribution
Example: If f(x) is a probability distribution
of a discrete random variable X, find the
value of c if
c
f ( x )  x , for x  1, 2, 3.
3
Sol:
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.
3
 f ( x)  f (1)  f (2)  f (3)  1
i 1
10
Property 2
Sol:
c
c
c
1 2 3
1   2   3  c  (   )
3
3
3
3 3 3
1
 c
 c2 1
2

11
 5  3 
  
1 1
f (1)  P( X  1)    
8
 
 2
12
Prob. Distributions & Densities - 2
Probability Distributions and Probability Densities
Probability Distribution
x
f(x) = P(X = x)
2
X=x
Is it a profitable insurance
premium?
f (x)
20/56
100
.1
.75
1
30/56
1
.25
.50
0
15/56
11
?
.65
.25
 5  3 
 

x  2  x 

f ( x) 
, for x  0, 1, 2.
8
 
 2
Probability line chart
-100 -1 11
Probability Distribution
Distribution: f (-100) = .1
f (-1) = .25,
f (11) = .65,
.65 , if

f ( x)  .25 , if
 .1 , if

13
Cumulative Distribution
Function
x  11
x  1
x  10014
Distribution Function
Example: (Toss a balanced coin 3 times)
X takes on number of heads occurs.
The cumulative distribution function (c.d.f.
or distribution function, d.f.) of a discrete
random variable is defined as
F ( x)  P( X  x)   f (t ), for    x  .
tx
• F( ) = 0, F() = 1
• If a < b, F(a) ≤ F(b) for any real numbers a, b.
x
0
1
2
3
f(x)
1/8
3/8
3/8
1/8
F(x)
1/8
4/8
7/8
1
15
F(x)
16
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, …
1
1/2
Example: (Toss a balanced coin 3 times)
X takes on number of heads occurs.
x
0
1
2
f(2) = F(2) – F(1) = 4/8 – 1/8 = 3/8
3
17
18
Prob. Distributions & Densities - 3
Probability Distributions and Probability Densities
Probability Distribution of the Sum
(Two Dice)
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)}
Let X be a random variable takes on the sum of
the two dice.
P(X = 6) = ?
P(X = 8) = ?
P(X ≤ 8) = ?
2. Sum:
S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
19
20
Probability Distribution of the Sum
(Two Dice)
Distribution Function of the Sum
(Two Dice)
21
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)
3.4 Continuous Random Variable &
Probability Density Function
Percent
22
f(x)
𝑏
A smooth curve that fit
the distribution
P(a ≤ X ≤ b) =
𝑓 𝑥 𝑑𝑥
𝑎
a b
x
for any real constants a and b with a ≤ b.
Density
function, f (x)
Test scores
10 20 30 40 50 60 70 80 90 100 110
23
Notes:
f(c) is the probability density at c.
f(c) ≠ P(X = c) = 0 for any continuous random variable.
24
Prob. Distributions & Densities - 4
Probability Distributions and Probability Densities
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.
Example: If X is a continuous random variable with
a p.d.f.
𝑘 ∙ 𝑒−3𝑥 ,
for 𝑥 > 0
𝑓 𝑥 =
0,
elsewhere
Find k.
Find P(0.5 < X < 1).
25
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
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
𝑑𝐹(𝑥)
𝑓 𝑥 =
𝑑𝑥
where the derivative exists.
𝑥
=
𝑓 𝑡 𝑑𝑡,
26
for − ∞ < 𝑥 < ∞
−∞
is called the distribution function, or the
cumulative distribution of X, and
• F(−∞) = 0, F(∞) = 1,
• If a < b, F(a) ≤ F(b).
27
Example: If X is a continuous random
variable with a p.d.f.
3𝑒−3𝑥 ,
for 𝑥 > 0
𝑓 𝑥 =
0,
elsewhere
Find the distribution function of X.
Use the d.f. above to find P(0.5 < X < 1).
29
28
Example: Find a probability density function for the
random variable whose distribution function is given
by
0,
for 𝑥 ≤ 0
𝐹 𝑥 = 𝑥,
for 0<𝑥 < 1
1,
for 𝑥 ≥ 1
and plot its graph.
Sol: The function is discontinuous at 0 and 1.
By Theorem 3.6
0,
for 𝑥 ≤ 0
𝑓 𝑥 = 1,
for 0<𝑥 < 1
0,
for 𝑥 ≥ 1
We can let f(x) = 0 at 0 and 1, then we’ll have
1,
for 0<𝑥 < 1
𝑓 𝑥 =
0,
elsewhere.
30
Prob. Distributions & Densities - 5
Probability Distributions and Probability Densities
Example: X is a random variable with the following d.f.,
0,
𝑥<0
𝑥/2,
0≤𝑥<1
1≤𝑥<2
𝐹 𝑥 = 3/4,
5/6,
2≤𝑥<3
1,
3 ≤ 𝑥.
A Mixed Distribution
What is its F(x)?
What is its f(x)?
F(x)
1
Find P(X < 2) =
P(X < 1/2) =
P(X = 1) =
P(X ≤ 3/2) =
P(X > 3/2) =
¾
½
¼
F(x)
1
½
0
1
2
3
x
0
0.5
1
31
32
3.5 Multivariate Distributions
Univariate
(one random variable)
Multivariate Distributions
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).
33
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.
Two balls are selected from a box containing 3 red
balls, 2 blue balls, and 4 green balls.
4 
 3  2 
  

x
y
2

x  y 





P ( x, y ) 
, x, y  0,1,2; 0  x  y  2.
9
 
 2
Sol: S = {(0,0), (0,1), (0,2), (1,0), (1,1), (2,0)}
 3  2  4 
   
0 0 2
1
P(0,0)      
6
9
 
2
 
34
y
35
2
1
0
1/36
2/9
1/6
0
Y
2
1/6
1/3
1
x
1
1/12
2
0
x
0
1
2
36
Prob. Distributions & Densities - 6
Probability Distributions and Probability Densities
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.
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
• f(x, y) ≥ 0, for each pair of values (x, y) within
its domain; ∞ < x < ∞;
•
= 1, where the double summation
extends over all possible pairs (x, y) within its
domain.
37
38
Example: In the example about drawing two
balls problem, find
• P(1 ≤ X ≤ 2, 0 ≤ Y ≤ 1) = ?
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.
1/6 + 1/3 + 1/12 = 7/12
y
• P(X ≤ 1, 0 ≤ Y ≤ 1) = ?
2
1
0
1/36
2/9
1/6
0
1/6
1/3
1
x
1/12
2
2/9 + 1/6 + 1/6 + 1/3 = 7/9
39
Example: In the last example about three colors
balls problem, find
F(1, 1) = ?
F(0.5, 1.2) = ?
F(1, 1) = ?
Definition 3.7. If X and Y are discrete random
variables, the function given by
F(x, y) = P(X ≤ x, Y ≤ y) =
 
s x
t y
40
f ( s, t ),
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.
y
41
2
1
0
1/36
2/9
1/6
0
Y
2
1/6
1/3
1
x
1
1/12
2
0
x
0
1
2
42
Prob. Distributions & Densities - 7
Probability Distributions and Probability Densities
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
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
• f(x, y) ≥ 0, for ∞ < x < ∞; ∞ < y < ∞,
P[( X , Y )  A]   f ( x, y )dxdy
A
•
for any region A in the xy-plane.

 

 
f ( x, y ) dxdy  1 .
43
44
Example: The following is the joint probability
density function of two continuous random variables
X and Y,
Example: The following is the joint probability
density function of two continuous random variables
X and Y,
3
x( x  y )

f ( x, y )   5

 0
3
x( x  y )

f ( x, y )   5

 0
for 0  x  1, 0  y  2
elsewhere.
1. Find P(0 < X < Y, Y < 1).
for 0  x  1, 0  y  2
elsewhere.
2. Find P(0 < X < 1/2, 1< Y < 2).
3. Find P(0 < X + Y < 1/2).
45
yy
Example: The following is the joint yprobability
density function of two continuous random variables
X and Y,
1
 y
f ( x, y )  

0
for 0  x  y, 0  y  1
46
Definition 3.9: If X and Y are continuous random
variables with joint probability density function f(x, y),
the function given by
F ( x, y )  P ( X  x, Y  y )  
y

x
 
f ( s, t )dsdt
for ∞< x < ∞; ∞< y < ∞, is called the joint
distribution function, or the joint cumulative
distribution of X and Y.
elsewhere.
Find P(X + Y > 1/2).
f ( x, y) 
47
2
F ( x, y)
xy
48
Prob. Distributions & Densities - 8
Probability Distributions and Probability Densities
F(x, y)
If x < 0 or y < 0, F(x, y) = 0
Example: If the joint probability density of X and Y
is given by
x y
f ( x, y)  
0
for 0  x  1, 0  y  1
elsewhere.
1
y
F(x, y) =

0



y
IV
y
0
y
0
1
I
II
0
x
1
49
F(x, y)
(II)
F(x, y) =
1

0

0
1
2
y
III
1
( s  t )dsdt
y ( y  1)
I
II
0
If x > 1 and y > 1,
1
x

1
2
( s  t )dsdt
x
( 12 x 2  xt ) dt
y
1
2
xt 2 ]|

1
2
xy 2 
x2 y 
0
1
2
xy ( x  y )
50
F(x, y) =
x ( x  1)
y
1
1 1
  (s  t )dsdt
0 0
1
 0
 12 xy ( x  y )

F ( x, y )   12 y ( y  1)
 1 x( x  1)
2
 1
51
Example: Find the joint probability density of the
two random variables X and Y whose joint
distribution function is given by
(1  e  x )(1  e  y )
F ( x, y)  
0
III
I
x
0
If 0 < x < 1 and y > 1,
0 0
x
1
0
 [ 12 x 2t 
1
2
II
( 12 s 2  ts ) | dt
1
(IV)

( s  t )dsdt
IV
(III)
F(x, y) =
0
0
IV
F(x, y)
If x > 1 and 0 < y < 1,
y
x
III
I
If 0 < x < 1 and 0 < y < 1,
find the joint distribution function of X and Y.
III
y
(I)
IV
II
x
1
for x  0, y  0
for 0  x  1, 0  y  1
for x  1, 0  y  1
for 0  x  1, y  1
for x  1, y  1
52
P(1 < X < 3, 1 < Y < 2) =
for x  0 and y  0
elsewhere
and also determine P(1 < X < 3, 1 < Y < 2).
Sol: For joint p.d.f.,
f ( x, y) 
2
F ( x, y)
xy
 e  x e  y  e  ( x  y ) , for x  0, y  0
and 0 elsewhere.
53
54
Prob. Distributions & Densities - 9
Probability Distributions and Probability Densities
y
Multivariate Discrete Distribution
(a, d)
d
(b, d)
The joint probability distribution of n discrete
random variables, X1, X2, X3, …, Xn, is
(b, c)
c
f ( x1 , x2 ,..., xn )  P( X 1  x1 , X 2  x2 ,..., X n  xn )
(a, c)
x
0
a
for each (x1, x2, x3, …, xn) in the range of the r.v.’s,
b
and the joint distribution function
F ( x1 , x2 ,..., xn )  P( X 1  x1 , X 2  x2 ,..., X n  xn )
P(a<X<b, c<Y<d)
for −∞ < x1< ∞, −∞ < x2 < ∞, …, −∞ < xn < ∞.
= F(b,d) F(b,c)  F(a,d)  F(a,c)
55
56
Multivariate Continuous Distribution
Example: If the joint probability density of three
random variables X ,Y, and Z is given by
 ( x  y) z

for x  1,2; y  1,2,3; and z  1,2
f ( x, y, z )   63

elsewhere
 0
find P(X = 2, Y + Z ≤ 3).
The joint probability density function of n
continuous random variables, X1, X2, X3, …, Xn, can
be define by a multivariate function f ( x1 , x2 ,..., xn ) ,
and the joint distribution function is
F ( x1 , x2 ,..., xn )  P( X 1  x1 , X 2  x2 ,..., X n  xn )
xn
x2

 
  ...

x1
f (t1 ,t2 ,..., tn )dt1dt 2 ...dt n
for −∞ < x1< ∞, −∞ < x2 < ∞, …, −∞ < xn < ∞,
and
n
f ( x1 , x2 ,..., xn ) 
57

F ( x1 , x2 ,..., xn ).
x1x2 ...xn
58
Two balls are selected from a box containing 3 red
balls, 2 blue balls, and 4 green balls.
Example: If the joint probability density function of
X1, X2, and X3 is given by
( x  x )e x3 for 0  x1  1, 0  x2  1, 0  x3
f ( x1 , x2 , x3 )   1 2
elsewhere.
 0
find P(0 < X1 < 1/2, 1/2 < X2 < 1, X3 < 1).
4 
 3  2 
  

x
y
2

x  y 





P ( x, y ) 
, x, y  0,1,2; 0  x  y  2.
9
 
 2
y
59
2
1
0
1/36
2/9
1/6
0
Y
2
1/6
1/3
1
x
1
1/12
2
0
x
0
1
2
60
Prob. Distributions & Densities - 10
Probability Distributions and Probability Densities
3.6 Marginal Distributions
y
5/12
2 1/36
1 2/9
0 1/6
0
1/2 1/12
1/36
1/6
7/18
1/3 1/12 7/12
1
2
x
y
Y
2
1
0
5/12
2 1/36
1 2/9
0 1/6
0
x
0
1
1/2 1/12
1/6
1/3 1/12
1
2
x
The Marginal Probability Distribution of X:
2
2
g ( x)   f ( x, y), for x  0, 1, 2.
y 0
g (0)  5 / 12, g (1)  1 / 2, g (2)  1 / 12.
61
y
2 1/36
1 2/9
0 1/6
0
62
Definition 3.11. If X and Y are discrete random
variables with joint probability distribution f(x, y),
the function given by
1/36
1/6
7/18
1/3 1/12 7/12
1
2
x
g ( x)   f ( x, y)
y
for each x in the space of X, is called the marginal
distribution of X. Correspondingly,
The Marginal Probability Distribution of Y:
h( y)   f ( x, y)
2
h( y)   f ( x, y), for y  0, 1, 2.
x
x 0
h(0)  7 / 12, h(1)  7 / 18, h(2)  1 / 36.
for each y in the space of Y, is called the marginal
distribution of Y.
63
Definition 3.11. If X and Y are continuous random
variables with joint probability density f(x, y), the
function given by
Example: Given the joint probability density
function of two continuous random variables X and Y,

g ( x) 
64
 f ( x, y)dy

for   x <  is called the marginal distribution of
X. Correspondingly,
2
 ( x  2 y),
f ( x, y )   3

0,

for 0  x  1, 0  y  1
elsewhere
find the marginal density of X and Y.

h( y) 
 f ( x, y)dx

for   y <  is called the marginal distribution of
Y.
65
66
Prob. Distributions & Densities - 11
Probability Distributions and Probability Densities
Example: Given the trivariate joint probability
density function random variables X1, X2, X3,
( x  x )e x3 , for 0  x1  1, 0  x2  1, x3  0
f ( x1 , x2 , x3 )   1 2
0,
elsewhere,

find the marginal density of X1.
Example: Given the trivariate joint probability
density function random variables X1, X2, X3,
( x  x )e x3 , for 0  x1  1, 0  x2  1, x3  0
f ( x1 , x2 , x3 )   1 2
0,
elsewhere

find the joint marginal density of X1 and X3.
67
Definition 3.11. If X and Y are discrete random
variables with joint probability distribution f(x, y),
the function given by
68
Definition 3.11. If X and Y are continuous random
variables with joint probability density f(x, y), the
function given by

g ( x)   f ( x, y)
g ( x) 
 f ( x, y)dy

y
for   x <  is called the marginal probability
density of X. Correspondingly,
for each x in the space of X, is called the marginal
distribution of X. Correspondingly,

h( y)   f ( x, y)
h( y) 
 f ( x, y)dx

x
for   y <  is called the marginal probability
density of Y.
for each y in the space of Y, is called the marginal
distribution of Y.
69
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
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
g ( x1 )  ... f ( x1 , x2 ,..., xn )
x2
g ( x1 )  


xn
for all x1 in the space of X1. The joint marginal
probability distribution of X1, X2, X3 is
...


f ( x1 , x2 ,..., xn )dx2 ...dxn
for all x1 in the space of X1. The joint marginal
probability density of X1, X2, X3 is
m( x1 , x2 , x3 )  ... f ( x1 , x2 ,..., xn )
x4
70
m( x1 , x2 , x3 )  


xn
...


f ( x1 , x2 ,..., xn )dx4 ...dxn
for all   x1 < ,   x2 < , …,   xn < .
for all x1 , x2, x3, is in the space of X1, X2, X3.
71
72
Prob. Distributions & Densities - 12
Probability Distributions and Probability Densities
Example: Given the joint distribution function of
two continuous random variables X and Y,
Joint Marginal Distribution Function

(1  e  x )(1  e  y ), for x  0, y  0
F ( x, y)  

elsewhere,
 0,
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 < .
2
2
find the marginal distribution of 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 < .

(1  e  x ), for x  0
G( x)  F ( x, )  

elsewhere.
0,
2
x 
y x
F ( x, y) 
  f (s, t )dsdt
G ( x) 

  f (s, t )dtds

73
74
3.7 Conditional Distribution
Example: Given the trivariate joint probability
density function random variables X1, X2, X3,
( x  x )e x3 , for 0  x1  1, 0  x2  1, x3  0
f ( x1 , x2 , x3 )   1 2
0,
elsewhere,

P( A | B) 
P ( A  B)
P ( B)
Discrete Case:
If A and B are events X = x and Y = y, then
find the marginal distribution of X1, G(x1).
P ( X  x, Y  y )
P(Y  y )
f ( x, y )

h( y )
P( X  x | Y  y ) 
75
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
f ( x | y) 
f ( x, y)
, h(y)  0
h( y)
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
w( y | x) 
f ( x, y)
, g x   0
g ( x)
is called the marginal conditional distribution of Y
given X = x.
77
76
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
f ( x | y) 
f ( x, y)
, h(y)  0
h( y)
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
f ( x, y)
w( y | x) 
g ( x)
, g x   0
is called the marginal conditional probability
density function of Y given X = x.
78
Prob. Distributions & Densities - 13
Probability Distributions and Probability Densities
5/12
2 1/36
y 1 2/9
0 1/6
0
1/2 1/12
1/36
1/6
7/18
1/3 1/12 7/12
1
2
x
Conditional probability distribution of X given Y = 1:
2/9
f (0 | 1) 
 4/7
7 / 18
1/ 6
f (1 | 1) 
 3/ 7
7 / 18
f ( 2 | 1)  0
5/12
2 1/36
1 2/9
0 1/6
0
79
1/2 1/12
5/12
2 1/36
y 1 2/9
0 1/6
0
1/2 1/12
1/36
1/6
7/18
1/3 1/12 7/12
1
2
x
Find P( X ≤ 1 | Y = 0 )
= P( X = 0 | Y = 0 ) + P( X = 1 | Y = 0 )
P( X  0, Y  0) P( X  1, Y  0)


P(Y  0)
P(Y  0)
1/ 6
1/ 3 6



7 / 12 7 / 12 7
Example: Given the joint probability density
function of two continuous random variables X and Y,
1/36
y
1/6
7/18
1/3 1/12 7/12
1
2
x
Find P( X ≤ 1 | Y ≤ 1 ) = ?
P ( X  1, Y  1)

P (Y  1)
2 / 9 1/ 6 1/ 6 1/ 3

7 / 18  7 / 12
2
 ( x  2 y ),
f ( x, y )   3

0,

for 0  x  1, 0  y  1,
elsewhere
find the marginal conditional density of X given
Y = y and use it to calculate P(X ≤ 1/2 | Y = 1/2).
81
2
 ( x  2 y ),
f ( x, y )   3

0,

80
for 0  x  1, 0  y  1,
82
f (x
elsewhere
1
2x  2 2x  2
)

2
1 2
3
1
The marginal conditional density of X given Y = y
is
1
h( y )  (1  4 y ), for 0  x  1,
3
2
( x  2 y)
f ( x, y ) 3
2x  4 y
f ( x | y) 


, for 0  x  1,
1
h( y )
1 4 y
(1  4 y )
3
and f ( x | y )  0 elsewhere.
83
P( X 
2
1
1
1
| Y  )   f ( x | )dx
2
2 0
2
1
2

0
2x  2
5
dx 
3
12
84
Prob. Distributions & Densities - 14
Probability Distributions and Probability Densities
Example: Given the joint probability density
function of two continuous random variables X and Y,
Can it be done in the following way?
1
1
P( X  | Y  ) 
2
2
1
1
,Y  )
2
2
1
P(Y  )
2
P( X 
4 xy,
f ( x, y)  
 0,
find the marginal density of X and Y and the
conditional p.d.f. of X given Y = y .
Can we do the following if both X and Y are
continuous random variables?
1
1
P( X  | Y  ) 
2
2
for 0  x  1, 0  y  1
elsewhere
1
1
,Y  )
2
2
1
P(Y  )
2
P( X 
85
More than two variables:
x1 , x2 , x3 , x4  f ( x1 , x2 , x3 , x4 )
p ( x1 | x2 , x3 , x4 ) 
f ( x1 , x2 , x3 , x4 )
, g ( x2 , x3 , x4 )  0.
g ( x2 , x3 , x4 )
q( x1 , x2 | x3 , x4 ) 
f ( x1 , x2 , x3 , x4 )
, m( x3 , x4 )  0.
m( x3 , x4 )
r ( x2 , x3 , x4 | x1 ) 
f ( x1 , x2 , x3 , x4 )
, l ( x1 )  0.
l ( x1 )
86
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).
87
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.
88
Example: Given the independent random variables
X1, X2, and X3, with the probability density functions
 e x1 ,
f1 ( x1 )  
0,
2e2 x2 ,
f 2 ( x2 )  
 0,
3e3 x3 ,
f 3 ( x3 )  
 0,
for x1  0
elsewhere,
for x2  0
elsewhere,
for x3  0
elsewhere,
find the joint density of X1, X2, and X3, and also
find P(X1 + X2 ≤ 1, X3 > 1).
89
90
Prob. Distributions & Densities - 15