Dr Sonnet Hng, USTH Topic 2

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TOPIC 2: Random Variables andProbability Distributions

Course: Probability and StatisticsLect. Dr Quang Hng/ Sonnet Nguyen

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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

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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.

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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

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Random Variables

} A random variable is a function that assigns a real number to each outcome in the sample space of a random experiment.

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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 =

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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.

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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.

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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.

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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 =

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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

= ==

=

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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=

=

= =

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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

= = = =

= = = = = =

= = = + = + =

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Example of Discrete Probability Distributions

} The probability function is thus given by Table 2-2.

} Table 2-2

1/41/21/4f(x)

210x

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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= = =

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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

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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

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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.

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Applications to Geometric Probability} Fig. 2-5

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Applications to Geometric Probability

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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

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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.