Probability Theory 9

Physics Helpline
L K Satapathy
Probability Theory 9

Physics Helpline
L K Satapathy
Random Variable : In a random experiment, a single real number may be
assigned to each outcome () of the experiment ( S) , which may
be different for different outcomes.
Definition: A random variable is a real valued function whose domain is
the sample space of a random experiment.
For tossing two coins S = { HH , HT , TH , TT }
Let X denote ‘the number of heads’ obtained. Then X is a random
variable whose value for the different outcomes are as follows:
X(HH) = 2 , X(HT) = 1 , X(TH) = 1 and X(TT) = 0
We can define more than one random variable on the same sample space.
Let Y denote ‘the number of heads – the number of tails’ obtained.
Then Y(HH) = 2 – 0 =2 , Y(HT) = 1 – 1 = 0 ,
Y(TH) = 1 – 1 = 0 and Y(TT) = 0 – 2 = – 2

Physics Helpline
L K Satapathy
Example: A person plays the game of tossing three coins. For each
head he is given Rs 3 by the organiser and for each tail , he has to give
Rs 2 to the organiser. Let X denote the amount gained by the person.
Show that X is a random variable and exhibit it as a function on the
sample space of the experiment.
Ans : For 3 coins, S = { HHH , HHT , HTH , HTT , THH , THT , TTH , TTT }
X = Amount gained = (3  number of heads) – (2  number of tails)
 X (HHH) = 33 – 20 = 9 , X(HHT) = X(HTH) = X(THH) = 32 – 21 = 4
X(HTT) = X(THT) = X(TTH) = 31 – 22 = – 1 , X(TTT) = 30 – 23 = – 6
 X is a real number whose value is defined for each outcome
and X takes a unique value for each outcome of the experiment
 X is a function whose domain = S and whose range = { – 6 , – 1 , 4 , 9 }

Physics Helpline
L K Satapathy
Probability distribution of a Random Variable :
A description of the values of a random variable X along with the corresponding
probabilities is called the probability distribution of the random variable X.
Consider the experiment of selecting 1 family out of 10 families 1 2 10, , . . . ,f f f
in such a manner that each family is equally likely to be selected. Let the families
have 3 , 4 , 3 , 2 , 5 , 4 , 3 , 6 , 4 , 5 members respectively.1 2 10, , . . . ,f f f
Let us select a family at random and note the number of members denoted by X .
 X is a random variable defined as follows:
1 2 3 4 5( ) 3 , ( ) 4 , ( ) 3 , ( ) 2 , ( ) 5X f X f X f X f X f    
6 7 8 9 10( ) 4 , ( ) 3 , ( ) 6 , ( ) 4 , ( ) 5X f X f X f X f X f    
 X = 2 for ( ) , X = 3 for ( ) , X = 4 for ( )4f 1 3 7, ,f f f
X = 5 for ( ) and X = 6 for ( )
2 6 9, ,f f f
5 10,f f 8f

Physics Helpline
L K Satapathy
Each family is equally likely to be selected .
 Probability of selecting each family (from a total of 10) = 1/10
X = 2 for 1 family  P(X = 2) = 1/10
X = 3 for 3 families  P(X = 3) = 3/10
X = 4 for 3 families  P(X = 4) = 3/10
X = 5 for 2 families  P(X = 5) = 2/10
X = 6 for 1 family  P(X = 6) = 1/10
X 2 3 4 5 6
P(X) 1/10 3/10 3/10 2/10 1/10
Probability distribution
We observe that
1 3 3 2 1 10 1
10 10 10 10 10 10
     

Physics Helpline
L K Satapathy
In general , if be the possible values of the random variable X1 2, , . . . , nx x x
X
P(X)
1x 2x 3x nx  
1p 2p 3p np  
1
0, 1,2,3,..,
1
i
n
i
i
p i n
and p

 

Where
and the probability of X taking the value be ,( )i iP X x p ix
then the probability distribution of X is described as follows:

Physics Helpline
L K Satapathy
Mean of a Random Variable :
It is the measure of the central tendency or the average value of a random variable.
Definition :
Let X be a random variable whose possible values 1 2, , . . . , nx x x
1 1 2 2
1
( ) . . . .
n
i i n n
i
E X x p x p x p x p

     
Then mean of X , denoted by  is the weighted average of the possible values of X ,
each value being weighted by its probability of occurrence.
It is also called the expectation of X , denoted by E(X).
In other words , the mean or expectation of a random variable X is the sum of the
products of all the possible values of X with their respective probabilities.
occur with probabilities respectively.1 2, , . . . , np p p
Then

Physics Helpline
L K Satapathy
Example : Let a pair of dice be thrown and the random variable X be the sum of the
numbers appearing on the two dice. Find the mean or expectation of X .
Answer : The sample space for throwing of two dice is
The random variable X ( sum of the numbers on the two dice ) can take any one of the
values 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 .
 S consists of 36 elementary events which are in the form of ordered pairs (x , y) ,
where each of x and y can take the values of 1 , 2 , 3 , 4 , 5 or 6 .
 Probability of each elementary event = 1/36 .
(1,1) , (1,2) , (1,3) , (1,4) , (1,5) , (1,6) ,
(2,1) , (2,2) , (2,3) , (2,4) , (2,5) , (2,6) ,
(3,1) , (3,2) , (3,3) , (3,4) , (3,5) , (3,6) ,
(4,1) , (4,2) , (4,3) , (4,4) , (4,5) , (4,6) ,
(5,1) , (5,2) , (5,3) , (5,4) , (5,5) , (5,6) ,
(6,1) , (6,2) , (6,3) , (6,4) , (6,5) , (6,6)
S =

Physics Helpline
L K Satapathy
  5( 8) {(2,6),(3,5),(4,4),(5,3),(6,2)}
36
P X P  
  4( 9) {(3,6),(4,5),(5,4),(6,3)}
36
P X P  
  3( 10) {(4,6),(5,5),(6,4)}
36
P X P  
  2( 11) {(5,6),(6,5)}
36
P X P  
  1( 12) {(6,6)}
36
P X P  
  1( 2) {(1,1)}
36
P X P  Now
  2( 3) {(1,2),(2,1)}
36
P X P  
  3( 4) {(1,3),(2,2),(3,1)}
36
P X P  
  4( 5) {(1,4),(2,3),(3,2),(4,1)}
36
P X P  
  5( 6) {(1,5),(2,4),(3,3),(4,2),(5,1)}
36
P X P  
  6( 7) {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}
36
P X P  

Physics Helpline
L K Satapathy
1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
( )P X
X 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 5( ) 2 3 4 5 6 7 8
36 36 36 36 36 36 36
E X              

4 3 2 19 10 11 12
36 36 36 36
       

2 6 12 20 30 42 40 36 30 22 12
36
         
252 7 [ ]
36
Ans 

Physics Helpline
L K Satapathy
Variance of a Random Variable :
It is the variability or spread in the values of a random variable.
Definition :
probabilities respectively.1 2( ), ( ), . . . , ( )np x p x p x
Let X be a random variable whose possible values occur with1 2, , . . . , nx x x
Let be the mean of X . Then the variance of X is defined as( )E X 
2 2
1
( ) ( ) . ( )
n
x i i
i
Var X x p x 

  
Standard Deviation of a Random Variable :
It is the non-negative square root of the variance of a random variable defined as
2
1
( ) ( ) . ( )
n
x i i
i
Var X x p x 

  

Physics Helpline
L K Satapathy
Alternative expression for Variance of a Random Variable :
2 2
1
( ) ( ) . ( )
n
x i i
i
Var X x p x 

  
2 2
1
( 2 ). ( )
n
i i i
i
x x p x 

  
2 2
1 1 1
. ( ) . ( ) 2 . ( )
n n n
i i i i i
i i i
x p x p x x p x 
  
    
2 2
1 1 1
. ( ) . ( ) 2 . . ( )
n n n
i i i i i
i i i
x p x p x x p x 
  
    
2 2 2
1
. ( ) 2
n
i i
i
x p x  

   1 1
[ ( ) 1 . ( ) ]
n n
i i i
i i
Since p x and x p x 
 
  
2 2
1
. ( )
n
i i
i
x p x 

 
2 2 2
( ) ( ) [ ( )]xor Var X E X E X    2 2
1
[ , ( ) . ( ) ]
n
i i
i
where E X x p x

 

Physics Helpline
L K Satapathy
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Probability Theory 9

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Probability Theory 9