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Probability Distributions and Probability Densities Prob. Distributions & Densities - 1 1 Probability Distributions and Probability Densities 2 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. 3 Discrete Random Variable Discrete Random Variable: A random variable assumes discrete values by chance. 4 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 5 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, … 6 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

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  • Probability Distributions and Probability Densities

    Prob. Distributions & Densities - 1

    1

    Probability Distributions and

    Probability Densities

    2

    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.

    3

    Discrete Random Variable

    Discrete Random Variable:

    A random variable

    assumes discrete values

    by chance.

    4

    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

    5

    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, …

    6

    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

    7

    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.

    8

    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.

    9

    Probability Distribution

    Example: Is f(x) a proper probability

    distribution of the random variable X?

    3. 1, 1, for ,3

    )( xx

    xf

    10

    Probability Distribution

    Example: Is f(x) a proper probability

    distribution of the random variable X?

    3. 2, 1, for ,3

    )( xx

    xf

    11

    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

    12

    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

    13

    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

    14

    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

    15

    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.

    16

    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

    18

    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

    19

    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)

    22

    Distribution Function of the Sum

    (Two Dice)

    23

    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)

    24

    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

    25

    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 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).

    28

    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.

    29

    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

  • Probability Distributions and Probability Densities

    Prob. Distributions & Densities - 6

    31

    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).

    34

    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 yt tsf ),,(

    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

    )( 20 2

    1

    2

    212

    21 xyyx )(2

    1 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

  • 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 ,)(),,( 32121321

    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 ,)(),,( 32121321

    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 ),...,,(...)( 2112

    for all x1 in the space of X1. The joint marginal

    probability distribution of X1, X2, X3 is

    nx

    n

    x

    xxxfxxxm ),...,,(...),,( 213214

    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 ,)(),,( 32121321

    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 ,)( 111

    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