probability 3

8/13/2019 Probability 3

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A random variable X on a sample space S is a function from S into

the set R of real numbers such that the preimage of every interval

of R is an event of R is an event of S.

Random Variable

If X is a variable whose possible values are numerical outcomes of a random

experiment then X is called as a random variable. There are two types of

random variables, discreteand continuous.

Discrete Random Variables

A discrete random variable is one which may tae on only a

countable number of distinct values such as !,",#,$,%,........

&iscrete random variables are usually counts.

'g. the number of children in a family,

the number of students attendance at a tutorial class,

the number of patients in a doctor(s surgery,

Probability Distribution

The probability distribution of a discrete random variable is a list

of probabilities associated with each of its possible values. It isalso sometimes called the probability function or the probability

mass function.

Example:

A pair of fair dice is tossed. Then

S ={(1,1), (1,2), .., (6,1), (6,2).,(6,6)}


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Let the rv X = a! (a,") . ie X (a,") = a! (a,")

Then X(s) = {1,2,#,$,%,6}

&(X=1) = p{(1,1)} = 1'#6 = f(1)

&(X=2) =&{(1,2), (2,2), (2,1)} Ths f(2)=#'#6

Siiar* f#=%'#6, f$=+'#+, f% = '#6, f6 = 11'#6

Ths -e can for a ta"e

!i 1 2 # $ % 6

f(!i) 1'#6 #'#6 %'#6 +'#6 '#6 11'#6

This is caed the distri"tion of X

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2 3 4 5 6

Y value

Pr

obability

)igure " * +robability distribution of X

Note The above probability distribution may also be described bythe probability histogram


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Distribution of a RV

-et X be a rv on a sample space S with finite image set, Xs/ 0 1x",

x#, x$, 2..xn3. -et the function f on Xs/ , denoted by fxi/ 0

+X0xi, i0",#2.n/ is called the distribution or probability function

of X.

x" x# 2. xn

fx"/ )x#/ )xn/

The distribution f satisfies the conditions

a/ fxi/ 4 ! and b/ =

=n

i

xif"

"/.

A rando varia"e X is said to "e discreteif the set X is finiteor countable.

o- et / "e defined as the s of (a,")

i.e. /(a,") = a0"

Then / is a aso rv. The sape space of / {2,#,$,%,6,+,,,1,11,12}

3ind the distri"tion of /

/i 2 # $ % 6 + 1 11 12
http://planetmath.org/encyclopedia/Finite.htmlhttp://planetmath.org/encyclopedia/Countable.htmlhttp://planetmath.org/encyclopedia/Finite.htmlhttp://planetmath.org/encyclopedia/Countable.html


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&(*i) 1'#6 2'#6 #'#6 $'#6 %'#6 6'#6 %'#6 $'#6 #'#6 2'#6 1'#6

4.

pro"1'#6 #'#6 6'#6 1'#6 1%'#6 21'#6 26'#6 #'#6 ##'#6 #%'#6 #6'#6

Distribution of Y

1

2

#

$

%

6

+

2 # $ % 6 + , 1 11 12

)igure # * +robability distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.

0.!

0."

1

2 3 4 5 6 ! " 10 11 12

Possible values of #$

%umulativeProb.

3i5re 1.1 4ative distri"tion of X


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Continuous Random Variables

A continuous random variableis one -hich taes an infinite n"er of possi"e vaes.4ontinos rando varia"es are sa* easreents.

75. 8ei5ht of first *ear stdents in 9o: the tie re;ired to rn a

A continuous random variable is not defined at specific values.

Instead, it is defined over an interval of values, and is represented

by the area under a curvein advanced mathematics, this is

nown as an integral/.

+robability density function of continuous uniform variable

'g. fx/ 0 "5b6a if a 7 x 7b

0 ! , otherwise

888 +lot 888

Show another distribution

Properties of a random variable (discrete /continuous) [or

distribution]

888 explain 88

S5T 0"# and 'X/0$59

Properties of (!)

". 'xpected value of a constant is e:ual to that constant. i.e. If c is a constant, 'c/ 0 c


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#. If Xand Yare random variables such that X 7 ; the 'X/ 7 ';/

$. 'X < ;/ 0 ' X/ < ';/

%. 'X < c/ 0 ' X/ < c

9. ' aX / 0 a'X/

Variance of a distribution #. )

&'en ( is )is*rete

./f.x/.#

i

# = XEx

i

i , -here 7(X) =

=## /. i

i

i xfx = 7(X2) > ?7(!)@2

(Tr* soe cass e!ercises)

&'en ( is *ontinuous

$+(, - /.#

ii xfx

#

Ths -e can for a ta"e

!i 1 2 # $ % 6

f(!i) 1'#6 #'#6 %'#6 +'#6 '#6 11'#6

E+(2

, - 12

/+136, 22

/+336, 32

/+536, 42

/+36, 52

/+"36, 62

/+1136,

- 0136 - 21."

E+(, - 4.4 +s'oe), 'us E+(2, - 1"."!


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$+(, - 21." 1".!! - 1.""

o *onsi)er t'e #$ Y - sum of t'e numbers +ab, in a pair of )i*e.

/i 2 # $ % 6 + 1 11 12

&(*i) 1'#6 2'#6 #'#6 $'#6 %'#6 6'#6 %'#6 $'#6 #'#6 2'#6 1'#6

E+Y2, - 22/+136, 32/+236, 77777.. 112/+236, 122/+136,

- 54.!

E+Y, -

$+Y, - 54.! 2 - 5.!

8t). )ev of Y - - 2.4

probability 3

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