Transcript No Slide Title

PART 2
Probability and Random
Variables
Huseyin Bilgekul
Eeng571 Probability and astochastic Processes
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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Chapter 4
Distribution Functions and Discrete
Random Variables
4.1 Random Variables
4.2 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
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4.1 Random Variables
Definition
Let S be the sample space of an experiment. A realvalued function X：SR 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)})
Sum, s 5
6
7
8
9
10 11 12
=
=
P(X=4)P(X
= P({(1,3),
(2,2), (3,1)}) =
and, similarly
s)
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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：SR
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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 realvalued 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)
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= 2,
Example 4.1 (Cont’d)
What are the probabilities of X = 0, 1, 2, 3 ?
Sol：
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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
1
S  {t : 11  t  11 }
2
by
ThenP( X  t )  0 for any
P( X  ( ,  )) 
 
11  11
1
2
subinterva l ( ,  ) of
X (t )  t.
t S
 2(    )
for
any
(11, 11 12 )
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Example 4.3
In 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
S  {T , NT , NNT , NNNT ,} by
X (
NNN
N T)  i



i 1
The set of all possible values of X is {1, 2, 3, …}
and
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Example 4.4
In 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.
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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 i1. Thus,
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Example 4.5
The 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
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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
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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)
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Properties of CDF
1.
2.
3.
4.
5.
P(X > a) = 1  F(a)
P(a < X  b) = F(b)  F(a)
P(X < a) = lim n F(a  1/n)  F(a)
P(X  a) = 1  F(a)
P(X = a) = F(a)  F(a)
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Example 4.7
The distribution function of a random variable X is given by
0
x
4


F ( x)   12
1 x 1
2
 12

1
x0
0  x 1
1 x  2
2 x3
x3
Compute the following quanties：
(a) P(X < 2)
(b) P(X = 2)
(d) P(X > 3/2)
(e) P(X = 5/2)
(c) P(1 X < 3)
(f) P(2<X  7)
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Example 4.8
For 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 / 4

Ans : F (t )  
3 / 4
1
t  0,
0  t  1,
1  t  2,
t  2.
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Example 4.9
Suppose 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：
t  10,
0

Ans : F (t )  2(t  10) 10  t  10.5,
1
t  10.5.

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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：
t 0
0
1 t 2
0  t 1
2
F (t )  
2
k
(
4
t

t
) 1 t  2


t  2.
1
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?
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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(x
i) = P(X = xi) and hence p(xi)  0 (i = 1, 2, 3, …)

(c)  p( x i )  1.
i 1
Also called probability function.
n 1
F (t )  P( X  t )   p( xi ), where xn 1  t  xn
i 1
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Example 4.11
In 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：
x  1,
0
1 / 36

4 / 36

Ans : F ( x)  9 / 36
16 / 36

25 / 36

1
1  x  2,
2 x3
3  x  4.
4  x  5,
5  x  6,
x  6.
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Example 4.12
Can a function of the form
c( 23 ) x x  1,2,3,...
p( x)  
elsewhere.
 0
be a probability mass function ?
Sol：
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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：
t  1,
0
Ans : F (t )  
n 1
1

(
1
/
2
)
n  1  t  n, n  2,3,4,.

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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
E ( X )   xp( x )
xA
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.
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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：
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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：
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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?
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Example 4.16 (Cont’d)
Sol：
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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?
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Example 4.17 (Cont’d)
Sol：
Ans: ~0.13
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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：
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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.
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Example 4.19 (Cont’d)
Sol：
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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：
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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
xn
chip
is
b/(w+b).
pn : the probabilit y that the n - th draw is white pn 
w  b  (n  1)c
Pf：xn : the number of white chips just before the n - th draw
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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 nn draws.
Show
that E(Xn) w= nw/(w+b).
k
nk
P( X n  k )  Ck p (1  p)
, where p 
Binomial Distri.
w

b
Pf： n
n
n
n!
p k (1  p) nk
E ( X n )   kP( X n  k )   kCkn p k (1  p) nk   k
k!(n  k )!
k 1
k 0
k 0
n
n
(n  1)!
 n
p k (1  p) nk  np Ckn11 p k 1 (1  p)( n1)( k 1)
k 1 (k  1)!(n  k )!
k 1
n 1
 np C
m 0
n 1
m
p (1  p)
m
( n 1)  m
n1
 np[ p  (1  p)]
nw
 np 
wb
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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：
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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
E[ g ( X )] variable
  g ( x ) pwith
( x)
xA
p( g ( X )  z )  p( X  g 1 ( z )) 
Pf：
 p ( x)
{ x:g ( x )  z }
E[ g ( X )] 
 zp( g ( X )  z )   z  p( x)
zg ( A )

zg ( A ) { x: g ( x )  z }
  zp( x)    g ( x) p( x)
zg ( A ) { x:g ( x )  z }
zg ( A ) { x:g ( x )  z }
  g ( x) p( x)
xA
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Corollary
Let X be a discrete random variable; g1, g2, …, gn be
real-valued functions, and let 1,  2, …,  n be real
numbers. Then
E[1 g1 ( X )   2 g2 ( X )     n gn ( X )]
 1 E[ g1 ( X )]   2 E[ g2 ( X )]     n E[ gn ( X )]
Pf：
E[1 g1 ( X )     n g n ( X )]  [1 g1 ( x)     n g n ( x)] p( x)
xA
 [1 g1 ( x) p( x)     n g n ( x) p( x)]
xA
 1  g1 ( x) p( x)     n  g n ( x) p( x)
xA
xA
 1E[ g1 ( X )]    n E[ gn ( X )]
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Example 4.23
The probability mass function of a discrete random
variable X is given by
 x 15 x  1,2,3,4,5,
p ( x)  
0 otherwise.
What is the expected value of X(6 X) ?
Sol：
Ans：7
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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
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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.
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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
Var ( X )  E[( X   ) 2 ]   ( x   ) 2 p( x )
xA
 X  E[( X   ) 2 ].
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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.
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Example 4.26 (Cont’d)
Sol：
Ans: E(B) = 0.25, E(K) = 0.25
Var(B) = 55.69, Var(K) = 1.6875
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Theorem 4.3
Var ( X )  E ( X 2 )  [ E ( X )]2
Pf：
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Example 4.27
What is the variance of the random variable X, the
outcome of rolling a fair die?
Sol：
Ans: 35/12
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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.
Assume there exists some k   such that p(k )  P( X  k )  0,
 contradiction There does not exist any k    p(k) >0.
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Theorem 4.5
Let X be a discrete random variable; then for
constants a and b we
have
that
2
Var (aX  b) a Var ( X ),
 aX b  a  X .
Pf：
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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
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Concentration
Definition
Let X and Y be two random variables and  be a
given point. If for all t > 0,
P( Y    t )  P( X    t )
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) .
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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(X1) (Xk)] The factorial kth moment of
X
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4.6 Standardized Random Variables
Definition
The random variable
standardized X.
X   ( X   ) / is
called the

X  1
E ( X )  E    E ( X )   0

   
X  1

Var ( X )  Var     2 Var ( X )  1.
   

Note: If X1=X+, then X1*= X *.
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