Upload
ctvd93
View
13
Download
4
Tags:
Embed Size (px)
DESCRIPTION
Random Variables
Citation preview
Dr Sonnet Hng, USTH Topic 2
1
TOPIC 2: Random Variables andProbability Distributions
Course: Probability and StatisticsLect. Dr Quang Hng/ Sonnet Nguyen
2
Contents
} Random Variables} Discrete Random Variables
Discrete Probability Distributions Discrete Cumulative Distribution Functions
} Continuous Random Variables Continuous Density Function Continuous Cumulative Distribution Functions
} Joint Distributions} Independent Variables} Change Variables} Functions of Random Variables} Convolution} Conditional Distributions} Geometric Probability
Dr Sonnet Hng, USTH Topic 2
2
3
Random Variables} We often summarize the outcome from a random
experiment by a simple number. In many of the examples of random experiments that we have considered, the sample space has been a description of possible outcomes.
} In some cases, descriptions of outcomes are sufficient, but in other cases, it is useful to associate a number with each outcome in the sample space.
} The variable that associates a number with the outcome of a random experiment is referred to as a random variable.
4
Random Variables
Suppose that to each point of a sample space we assign a number. We then have a defined on tfunction
random variablhe sample
spac esto
e. This function ichastic variable
s called a (or ) or more precisely a
(stochastic function). It is usually denoted by a capital letter such as or . In general, a random variable has some specified physical, geometrical, or oth
random f
er signi
un
fi
ctio
ca
n
nce.
X Y
Dr Sonnet Hng, USTH Topic 2
3
5
Random Variables
} A random variable is a function that assigns a real number to each outcome in the sample space of a random experiment.
6
Random Variables
Example 2.1
Let represent
Suppose that
the number of
a coin is tossed twice so th
heads that can come
at the sample space is { , , , }.
. With each sample point we can associate a nu r
pmb
ue
S HH HT TH TTX
=
for as shown in Table 2-1.Thus, for example, in the case of (i.e., 2 heads),
= 2, while for (1 head), 1. It follows that is a random variable.
X
HH X
X
HX T =
Dr Sonnet Hng, USTH Topic 2
4
7
Random Variables
Table 2-1
0112X
TTTHHTHHSample Point
Note that many other random variables could also be defined on this sample space, for example, the square of the number of heads, the number of heads minus the number of tails, etc.
8
Discrete and Continuous Random Variables
discrete random vA random variable that takes on a or number of values is called a
A random variable that take
finite countably infinite
noncountably infinites on a number of values is
ar
ca
iable.
lled a .
A is a random variable with an interval (eithe
nondiscrete random variable
continuous random r finite or infinite) of real numbers for its
varia ra
blenge.
Dr Sonnet Hng, USTH Topic 2
5
9
Discrete and Continuous Random Variables
Examples of discrete random variables: number of scratches on a surface, proportion of defective parts among 1000 tested, number of transmitted bits received in error, etc.Examples of continuous random variables: electrical current, length, pressure, temperature, time, voltage, weight, etc.
10
Discrete and Continuous Random Variables
A voice communication system for a business contains 84 external lines. At a particular time, the system is observed, and some of the lines are being used. Let the random variable de
Exampl
not e
e:
e thX number of lines in use. Then, can assume any of the integer values 0 through 84. When the system is observed, if 10 lines are in use, 10.
Xx =
Dr Sonnet Hng, USTH Topic 2
6
11
Discrete Probability Distributions
1 2 3
Let be a discrete random variable, and suppose that the possible values that it can assume are given by
, , , ..., arranged in some order. Suppose also that these values are assumed with proba
X
x x x
probability fun
bilities given by ( = ) = ( ), 1, 2, . . . (1)
It is convenient to introduce the , actionprobability distribution probabil
lsoity
referred to as or
, given by mass
functi o n
k kP X x f x k =
( ) ( ).For , this reduces to Eq. (1) while for other values of , ( ) 0.
k
P X x f xx x
x f x
= ==
=
12
Properties of Discrete Probability Distributions
1
In general, ( ) is a if1) ( ) 0,2) ( ) 1 where the sum is taken over all possible values of .
For a discrete random variable with possibl
probability fun
e valuespr
,... ,o
cti
babil
n
o
a
x
n
f xf x
f x x
X x x
=
1
is a function such that1) (
ity function) 0,
2) ( ) 1,
3) ( ) ( ).
in
ii
i i
f x
f x
f x P X x=
=
= =
Dr Sonnet Hng, USTH Topic 2
7
13
Example of Discrete Probability Distributions
Find the probability function corresponding to the random variabExample 2.2
representing the number of heads facing up after tossing a coin twi
le ce.
X
Solution. Assuming that the coin is fair, we have1 1 1 1( ) , ( ) , ( ) , ( )4 4 4 4
Then1 1( 0) ( ) , ( 2) ( ) ,4 4
1 1 1( 1) ( ) ( ) ( ) .4 4 2
P HH P HT P TH P TT
P X P TT P X P HH
P X P HT TH P HT P TH
= = = =
= = = = = =
= = = + = + =
14
Example of Discrete Probability Distributions
} The probability function is thus given by Table 2-2.
} Table 2-2
1/41/21/4f(x)
210x
Dr Sonnet Hng, USTH Topic 2
8
15
Distribution Functions for Random Variables
The , or briefly the , for a random variable is defined by
cumulative distributi
( ) ( ,where is any real number, i.e., .
on function (cdf)distribution fu
)nction
F x P XX
x xx
-
=
= =
= = =
1
here ( , ) ( , ) is the joint probability function and ( ) is the marginal probability function for .
f x y P X x Y yf x X= = =
60
Conditional Distributions
1
2
conditional probability function of Y give
We define( , ) ( | )( )
and call it the . Similarly, the conditional probability function of given is
( , ) ( | )( )
W
n X
f x yf y xf x
X Yf x yf x yf y
=
=
1
2
e shall sometimes denote ( | ) and ( | ) by ( | )and ( | ), respectively
f x y f y x f x yf y x
Dr Sonnet Hng, USTH Topic 2
31
61
Conditional Distributions
1
These ideas are easily extended to the case where , are continuous random variables. For example, the
is( , )
conditional
( | )( )
where ( , ) is the joi
density function of Y given X
X Y
f x yf y xf x
f x y
=
1
nt density function of and , and ( ) is the marginal density function of
X Yf x X
62
Conditional Distributions
Using the conditional density function we can, for example, find that the probability of being between and given that is
( | ) ( | )
Generalizations of these resul
d
c
Y c dx X x dx
P c Y d x X x dx f y x dy
< < +
< < < < + = ts are also available.
Dr Sonnet Hng, USTH Topic 2
32
63
Applications to Geometric Probability} Fig. 2-5
64
Applications to Geometric Probability
1
Suppose that we have a target in the form of a plane region of area and a portion of it with area , as in Fig. 2-5. Then it is reasonable to suppose that the probability of hitting the region of a
K K
1 1
11
rea is proportional to . We thus define
(hitting region of area ) =
where it is assumed that the probability of hitting the target is 1. Other assumptions can of course be made. For
K KKP KK
example, there could be less probability of hitting outer areas. The type of assumption used defines the probability distribution function.