Discrete R.V. 4-1

Chapter 4Distribution Functions and Discrete Random Variables

4.1 Random Variables4.2 Distribution Functions

4.3 Discrete Random Variables

4.4 Expectations of Discrete Random Variables

4.5 Variances and Moments of Discrete

Random Variables

4.6 Standardized Random Variables

Discrete R.V. 4-2

4.1 Random VariablesDefinition

Let S be the sample space of an experiment. A real-valued function X ： SR is called a random variable of the experiment if, for each interval I R, { s ： X(s) I } is an event.

Example ： If in rolling two fair dice, X is the sum, then X can only assume the values 2, 3, 4, …, 12 with the following probabilities ：

P(X=2) = P({(1,1)}) = , P(X=3) = P({(1,2), (2,1)}) =

P(X=4) = P({(1,3), (2,2), (3,1)}) = and, similarly

Sum, s 5 6 7 8 9 10 11 12

P(X = s)

Discrete R.V. 4-3

Another Definition Definition A random variable X is a process of assigning

a number X(s) to every outcome s of an experiment. The resulting function must satisfy the following two conditions but is otherwise arbitrary ：

1. The set {X x} is an event for every x. 2. The probabilities of the events {X = } and {X = -}

equal 0: P{X = } = 0, P{X = -} = 0. P.S. X(s) is a real-valued function X ： SR

Discrete R.V. 4-4

Example 4.1 Suppose that 3 cards are drawn from an

ordinary deck of 52 cards, one by one, at random and with replacement.

Let X be the number of spades drawn; then X is a random variable.

If an outcome of spades is denoted by s, and other outcomes are represented by t, then X is a real-valued function defined on the sample space

S={(s,s,s), (t,s,s), (s,t,s), (s,s,t), (t,t,s), (t,s,t), (s,t,t), (t,t,t)} X(s,s,s) = 3, X(t,s,s) = X(s,s,t) = X(s,t,s) = 2, X(t,t,s) = X(s,t,t) = X(t,s,t) = 1, X(t,t,t) = 0,

Discrete R.V. 4-5

Example 4.1 (Cont’d)

What are the probabilities of X = 0, 1, 2, 3 ? Sol ：

Discrete R.V. 4-6

Example 4.2 A bus stops at a station every day at some

random time between 11:00 AM and 11:30 AM. If X is the actual arrival time of the bus, X is a random variable. It is defined on the sample space

Then

.)(by }2

11111:{ ttXttS

SttXP any for 0)(

)11 ,11( of ),( lsubinterva

any for )(21111

)),((

21

21

XP

Discrete R.V. 4-7

Example 4.3In the United States, the number of twin births is approximately 1 in 90. Let X be the number of births in a certain hospital until the first twins are born. X is a random variable. Denote twin births by T and single births by N. Then X is a real-valued function defined on the sample space

The set of all possible values of X is {1, 2, 3, …} and

iTNNNNXNNNTNNTNTTSi

) (by },,,,{1

Discrete R.V. 4-8

Example 4.4In a certain country, the draft-status priorities of eligible men are determined according to their birthdays. Numbers 1 to 366 are assigned to men with birthdays on Jan 1 to Dec 31. Then numbers are selected at random, one by one and without replacement, from 1 to 366 until all of them are chosen. Those with birthdays corresponding to the 1st number drawn would have the highest draft priority, those with birthdays corresponding to the 2nd number drawn have the 2nd-highest priority, and so on.

Discrete R.V. 4-9

Example 4.4 (Cont’d)Let X be the largest of the first 10 numbers selected. Then X is a random variable that assume the values 10, 11, 12, …, 366. The event X = i occurs if the largest number among the first 10 is i, that is, if one of the first 10 numbers is i and the other 9 are from 1 through i1. Thus,

Discrete R.V. 4-10

Example 4.5The diameter of the metal disk manufactured by a factory is a random number between 4 and 4.5 . What is the probability that the area of such a flat disk chosen at random is at least 4.41 ? Sol ：

Ans: 3/5

Discrete R.V. 4-11

Example 4.6

A random number is selected from the interval (0, /2). What is the probability that its sine is greater than its cosine?

Sol ：

Ans: 1/2

Discrete R.V. 4-12

4.2 Distribution Functions

Definition If X is a random variable, then the function F

defined on (, ) by F(t)=P(X t) is called the distribution function or cumulative distribution function (CDF) of X.

Properties 1. F is nondecreasing. 2. lim t F(t) = 1.

3. lim t F(t) = 0.

4. F is right continuous. F(t+)=F(t)

Discrete R.V. 4-13

Properties of CDF

1. P(X > a) = 1 F(a)

2. P(a < X b) = F(b) F(a)

3. P(X < a) = lim n F(a 1/n) F(a)

4. P(X a) = 1 F(a)5. P(X = a) = F(a) F(a)

Discrete R.V. 4-14

Example 4.7

The distribution function of a random variable X is given by

Compute the following quanties ：(a) P(X < 2) (b) P(X = 2) (c) P(1 X < 3)(d) P(X > 3/2) (e) P(X = 5/2) (f) P(2<X 7)

31

32

21

10

00

)(

21

121

21

4

x

xx

x

x

x

xF

x

Discrete R.V. 4-15

Example 4.8For the experiment of flipping a fair coin twice, let X be the number of tails and calculate F(t), the distribution function of X, and then sketch its graph. Sol ：

0 1 2t

)(tF

14/32/1

4/1

3

Discrete R.V. 4-16

Example 4.9Suppose that a bus arrives at a station every day between 10:00 Am and 10:30 AM, at random. Let X be the arrival time; find the distribution function of X, F(t), and then sketch its graph. Sol ：

0 10 5.10t

)(tF

1

11

Discrete R.V. 4-17

Example 4.10 The sales of a convenience store on a randomly

selected day are X thousand dollars, where X is a random variable with a distribution function of the following form ：

Suppose that this convenience store’s total sales on any given day are less than $2000.(a)Find the value of k.(b)Let A and B be the events that tomorrow the

store’s total sales are between 500 and 1500 dollars, and over 1000 dollars, respectively. Find P(A) and P(B).

(c) Are A and B independent events?

.21

21)4(

10

00

)( 2

221

t

tttk

tt

t

tF

Discrete R.V. 4-18

4.3 Discrete Random Variables

Definition

The probability mass function p of a discrete random variable X whose set of possible values is {x1, x2, x3, …} is a function from R to R that satisfies the following properties.

(a) p(x) = 0 if x {x1, x2, x3, …}

(b) p(xi) = P(X = xi) and hence p(xi) 0 (i = 1, 2, 3, …)(c) Also called probability function.

1

.1)(i

ixp

nn

n

ii xtxxptXPtF

1

1

1

where,)()()(

Discrete R.V. 4-19

Example 4.11In the experiment of rolling a balanced die twice, let X be the maximum of the two numbers obtained. Determine and sketch the probability mass function and the distribution function of X.Sol ：

)4()4( XPp )})4,4(),3,4(),4,3(),2,4(),4,2(),1,4(),4,1({(P 36/7

)5()5( XPp )})5,5(),4,5(),5,4(),3,5(),5,3(),2,5(),5,2(),1,5(),5,1({(P 36/9)6()6( XPp

)})6,6(),5,6(),6,5(),4,6(),6,4(),3,6(),6,3(),2,6(),6,2(),1,6(),6,1({(P 36/11

Discrete R.V. 4-20

Example 4.11

.61

,6536/25

,5436/16

.4336/9

3236/4

,2136/1

,10

x

x

x

x

x

x

x

)()( xXPxF

,1x)1( XP

)1(p ,21 x)2()1( pp ,32 x

)3()2()1( ppp ,43 x)4()3()2()1( pppp ,54 x

)5()4()3()2()1( ppppp ,65 x)6()5()4()3()2()1( pppppp ,6x

0 1 2x

)(tF

36/9

36/4

36/1

3 4 5 6

1

36/25

36/16

Discrete R.V. 4-21

Example 4.12

Can a function of the form

be a probability mass function ?Sol ：

elsewhere.0

,...3,2,1)()( 3

2 xcxp

x

Discrete R.V. 4-22

Example 4.13 Let X be the number of births in a hospital until the first girl born. Determine the probability mass function and the distribution function of X. Assume that the probability is 1/2 that a baby born is a girl. Sol ：

.,4,3,2,1)2/1(1

,10)(: 1 nntn

ttFAns n

Discrete R.V. 4-23

4.4 Expectations Discrete R.V.Definition

The expected value of a discrete random variable X with the set of possible values A and probability mass function p(x) is defined by

We say that E(X) exists if this sum converges absolutely.

The expected value of a random variable X is also called the mathematical expectation, or mean, or simply expectation of X.

Ax

xxpXE )()(

Discrete R.V. 4-24

Example 4.14 We flip a fair coin twice and let X be the number of heads obtained. What is the expected value of X ?Sol ：

Discrete R.V. 4-25

Example 4.15 We write the numbers a1, a2, a3, …, an on n identical balls and mix them in a box. What is the expected value of a ball selected at random ?Sol ：

Discrete R.V. 4-26

Example 4.16 A college mathematics department sends 8 to 12 professors to the annual meeting of the American Mathematical Society, which lasts five days. The hotel at which the conference is held offers a bargain rate of a dollars per day per person if reservations are made 45 or more days in advance, but charges a cancellation fee of 2a dollars per person. The department is not certain how many professors will go. However, from past experience it is known that the probability of the attendance of i professors is 1/5 for i = 8, 9, 10,11 and 12. If the regular rate of the hotel is 2a dollars per day per person, should the department make any reservations? If so, how many?

Discrete R.V. 4-27

Example 4.16 (Cont’d) Sol ：

:nreservatio No aaaaaaXE 100)1201101009080(5

1)(

:nsreservatio 8 aaaaaaXE 60)8070605040(5

1)(

:nsreservatio 9 aaaaaaXE 4.56)7565554542(5

1)(

:nsreservatio 10 aaaaaaXE 2.54)7060504744(5

1)(

:nsreservatio 11 aaaaaaXE 4.53)6555524946(5

1)(

:nsreservatio 12 aaaaaaXE 54)6057545148(5

1)(

Discrete R.V. 4-28

Example 4.17 In the lottery of a certain state, players pick six different integers between 1 and 49, the order of selection being irrelevant. The lottery commission then selects six of these numbers at random as the winning numbers. A player wins the grand prize of $1,200,000 if all six numbers that he has selected match the winning numbers. He wins the 2nd and 3rd prizes of $800 and $35, respectively. What is the expected value of the amount a player wins in one game?

Discrete R.V. 4-29

Example 4.17 (Cont’d)Sol ：

Ans: ~0.13

Discrete R.V. 4-30

Example 4.18 (St. Petersburg Paradox)

In a game, the player flips a fair coin successively until he gets a head. If this occurs on the k-th flip, the player win 2k dollars.

How much should a person, who is willing to play a fair game, pay?

Sol ：

Discrete R.V. 4-31

Example 4.19 Let X0 be the amount of rain that will fall in the

United States on the next Christmas day. For n > 0, let Xn be the amount of rain that will fall in the United States on Christmas n years later.

Let N be the smallest number of years that elapse before we get a Christmas rainfall greater than X0. Suppose that P(Xi = Xj) = 0 if i j, the events concerning the amount of rain on Christmas days of different years are all independent, and the Xn’s are identically distributed. Find the expected value of N.

Discrete R.V. 4-32

Example 4.19 (Cont’d) Sol ：

) , , , ,()( 0302010 nXXXXXXXXPnNP

03210 ) , , , , ,max( XXXXXXP n

Discrete R.V. 4-33

Example 4.20 The tanks of a country’s army are numbered 1 to N. In a

war this country loses n random tanks to the enemy, who discovers that the captured tanks are numbered. If X1, X2,…, Xn are the numbers of the captured tanks, what is E(max Xi) ? How can the enemy use E(max Xi) to find an estimate of N, the total number of this country’s tanks?

Sol ：

)}{max(1

kXP ini

N

nki

nii kXPkXE )}{max()(max

1

nk where

N

nkNn

kn

C

Ck

11

N

nkNn nkn

kk

C )!()!1(

)!1(1

N

nkNn nkn

k

C

n

)!(!

!

N

nk

knN

n

CC

n

N

nk

kn

knN

n

CCC

n)( 1

11 N

n

Nn

C

nC 11

1)(max1

iXEn

nN

Discrete R.V. 4-34

Example 4.22 (Polya’s Urn Model)

An urn contains w white and b blue chips. A chip is drawn at random and then is returned to the urn along with c > 0 chips of the same color. Prove that if n = 2, 3, 4, …, such experiments are made, then at each draw the probability of a white chip is still w/(w+b). and the probability of a blue chip is b/(w+b).

Pf ： whiteis drawth - y that theprobabilit the: npn

drawth - thebeforejust chips whiteofnumber the: nxn

cnbw

xp n

n )1(

cnbw

xp

cnbw

cxpp n

nn

nn )1()1(

)1(1

11

1

cnbw

xcp nn

)1(11

111

)1(

])2([

nnn p

cnbw

pcnbwcp1p

bw

w

Discrete R.V. 4-35

Example 4.21 An urn contains w white and b blue chips. A chip is

drawn at random and then is returned to the urn along with c > 0 chips of the same color. This experiment is then repeated successively. Let Xn be the number of white chips drawn during the first n draws. Show that E(Xn) = nw/(w+b).

Pf ： bw

wpppCkXP knkn

kn where,)1()(

n

k

knknk

n

knn ppkCkXkPXE

00

)1()()( knkn

k

ppknk

nk

)1()!(!

!

1

n

k

knk ppknk

nn

1

)1()!()!1(

)!1(

n

k

knknk ppCnp

1

)1()1(111 )1(

1

0

)1(1 )1(n

m

mnmnm ppCnp 1)]1([ nppnp

bw

nwnp

Binomial Distri.

Discrete R.V. 4-36

Theorem 4.1 If X is a constant random variable, that is, if P(X

= c) = 1 for a constant c, then E(X) = c. Pf ：

Discrete R.V. 4-37

Theorem 4.2 Let X be a discrete random variable with set of

possible values A and probability mass function p(x), and let g be a real-valued function. Then g(X) is a random variable with

Pf ：

Ax

xpxgXgE )()()]([

})(:{

1 )())(())(()(zxgx

xpzgXPzXgPzp

Ax

Agz zxgxAgz zxgx

zxgxAgzAgz

xpxg

xpxgxzp

xpzzzpZEXgE

)()(

)()()(

)()()()]([

)( })(:{)( })(:{

})(:{)()(

)(XgZ

Discrete R.V. 4-38

Corollary Let X be a discrete random variable; g1, g2, …, gn

be real-valued functions, and let 1, 2, …, n be real numbers. Then

Pf ：

)]([)]([)]([

)]()()([

2211

2211

XgEXgEXgE

XgXgXgE

nn

nn

Ax

nnnn xpxgxgXgXgE )()]()([)]()([ 1111

Ax

nn xpxgxpxg )]()()()([ 11

Ax

nnAx

xpxgxpxg )()()()(11

)]([)]([ 11 XgEXgE nn

Discrete R.V. 4-39

Example 4.23 The probability mass function of a discrete random

variable X is given by

What is the expected value of X(6 X) ?

Sol ：

otherwise.0

,5,4,3,2,1)( 15 x

xpx

Ans ： 7

Discrete R.V. 4-40

Example 4.24 A box contains 10 disks of radii 1, 2, …, and 10,

respectively. What is the expected value of the area of a disk selected at random from this box?

Sol ：

Ans:38.5

Discrete R.V. 4-41

Example 4.25 (Investment) Let X be the amount paid to purchase an asset, and

let Y be the amount received from the sale of the same asset. Putting fixed-income securities aside, the ratio Y/X is a random variable called the total return and is denoted by R. Obviously, Y = RX. The ratio r = (Y X)/X is a random variable called the rate of return. Clearly, r = (Y / X) – 1 = R – 1, or R = 1 + r.

Let X be the total investment. Suppose that the portfolio of the investor consists of a total of n financial assets. Let wi be the fraction of investment in the i-th financial asset. Then Xi = wiX is the amount invested in the i-th financial asset, and wi is called the weight of asset i.

Discrete R.V. 4-42

4.5 Variances and Moments of Discrete R.V. Definition

Let X be a discrete random variable with a set of possible values A and probability mass function p(x), and E(X) = . Then Var(X) and X, called the variance and the standard deviation of X, respectively, are defined by

.])[(

)()(])[()(

2

22

XE

xpxXEXVar

X

Ax

Discrete R.V. 4-43

Example 4.26 Two games ： Bolita and Keno.

To play Bolita, you buy a ticket for $1, draws a ball at random from a box of 100 balls numbered 1 to 100. If the ball draw matches the number on your ticket, you win $75; otherwise, you lose.

To play Keno, you bet $1 on a single number that has a 25% chance to win. If you win, they will return you dollar plus two dollars more; other, they keep the dollar.

Let B and K be the amounts that you gain in one play of Bolita and Keno, respectively. Find the means and variances for B and K.

Discrete R.V. 4-44

Example 4.26 (Cont’d)Sol ：

Ans: E(B) = 0.25, E(K) = 0.25 Var(B) = 55.69, Var(K) = 1.6875

])[()( 2 BEBVar100

99)25.01(

100

1)25.074( 22

69.55

])[()( 2 KEKVar 75.0)25.01(25.0)25.02( 22

6875.1

Discrete R.V. 4-45

Theorem 4.3

Pf ：

22 )]([)()( XEXEXVar

Discrete R.V. 4-46

Example 4.27 What is the variance of the random variable

X, the outcome of rolling a fair die?Sol ：

Ans: 35/12

Discrete R.V. 4-47

Theorem 4.4 Let X be a discrete random variable with the

set of possible values A and mean . Then Var(X) = 0 if and only if X is a constant with probability 1.

Pf ： Prove it by contradiction. ,0)()(such that some exists thereAssume kXPkpk

contradiction There does not exist any k p(k) >0.

Discrete R.V. 4-48

Theorem 4.5 Let X be a discrete random variable; then for

constants a and b we have that

Pf ：

.

),()( 2

XbaX a

XVarabaXVar

Discrete R.V. 4-49

Example 4.28 Suppose that, for a discrete random variable

X, E(X) = 2 and E[X(X 4)] = 5. Find the variance and the standard deviation of 4X +12.

Sol ：

Ans: Var( 4X+12) = 144

Discrete R.V. 4-50

Concentration Definition Let X and Y be two random variables and be a

given point. If for all t > 0,

Then we say that X is more concentrated about than is Y.

Theorem 4.6 Suppose that X and Y are two random variables

with E(X) = E(Y) = . If X is more concentrated about than is Y, then Var(X) Var(Y) .

)()( tXPtYP

Discrete R.V. 4-51

Moments Definition E[g(X)] Definition E(Xn) The nth moment of X

E(|X|r) The rth absolutemoment of X

E(X c) The first moment of X about c

E[(X c)n] The nth moment of X about c

E[(X )n] The nth central moment of X about E[X(X1)‧‧ ‧(Xk)] The factorial kth moment of X

Remark 4.2 ： The existence of higher moments implies the existence of lower moments.

Discrete R.V. 4-52

4.6 Standardized Random Variables

Definition The random variable is called

the standardized X.

Note: If X1=X+, then X1

*= X *.

/)( XX

.1)(1

)(

0)(1

)(

2

XVarX

VarXVar

XEX

EXE