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UNIT – I
PROBABILITY
Introduction : The theory of probability was introduced by an Italian mathematician Galileo. And later it was
developed by Pascal. Probability is an important tool in the areas of Engineering, computers, physics, social,
Biological, Business and management sciences.
Random Experiment : An experiment which is conducted a large number of times under certain identical conditions
and all the outcomes of the experiment are pre fixed, then such experiment is known as “ Random experiment”.
EX : Tossing a coin, rolling a dice, selecting a student from a class, etc.,
Trail : Conducting a random experiment only once to find out the results is known as trail.
Outcome : The results of a random experiment is known as an outcome.
Ex : T, H are the outcomes in tossing a coin.
Sample Space : The set of all possible outcomes of a random experiment is known as sample space. Sample space is
denoted by ‘ S’. Each element of the sample space is called sample point.
Event : The outcomes or set of outcomes of the random experiment are known as an event. In other words , Subset
of the sample space is known as an event.
Mutually exclusive events : If the happening of an event prevents the happening of all other events , then such
events are known as “ mutually exclusive events “.
Ex : In tossing a coin getting head and getting tail are mutually exclusive.
Equally likely events : Events are said to be equally likely if and only if each and every event has equal chance of
happening.
Independent events : Events are said to be independent , if and only if the happening of an event does not depend
upon the happening of other event.
Probability :
Probability is defined in Three types they are :
- Mathematical or classical definition of probability.
- Statistical definition of probability.
- Axiomatic definition of probability.
Mathematical Definition : In a random experiment there are ‘ n ‘ exhaustive mutually exclusive events with ‘ m ‘
favorable cases of existing an event ’ E’ then the probability of happening of the event ‘ E ‘ is defined as
n
m
Total
P E
noofpossiblecases
No.offavorablecases ( )
Here m nand 0 P(E) 1
Statistical Definition : If a random experiment is conducted for a large number if times then the limit of the ratio
of number of happenings of the event to the number of trails is known as the probability of getting an event ‘ E’.
i.e., n
m P E lt n
Where m= no of happenings of the events ‘ E ‘
n= No of trails.
Axiomatic Definition of Probability : Axiomatic definition of probability associated with the sample space S. the
sample space is the set of all possible outcomes of the random experiment.
If we consider an event ‘A’ from a sample space S. Then the happening of the event must be as follows.
(i) P( A) 0
(ii) 0 P(A) 1
(iii) P(^ A)^1
Addition Theorem of probability ( Two events ) :
Statement : If A and B are any 2 events then prove that P(A B) = P(A)+P(B)-P(A B)
Proof : From the Venn diagram,
LetA ( A B ) (A B)
c
Taking probability on both sides
P( A) P( A B ) (A B)
c
P( A) P(AB )P(AB) 1
c
LetB ( B A ) (A B)
c
Taking probability on both sides
P( A) P( B A) (A B)
c
P( A) P(BA )P(AB) 2
c
Consider ( ) ( ) ( )
c c A B AB AB BA
Taking probability on both sides
( ) ( ) ( ) ( )
c c P AB P AB AB BA
( ) ( ) ( ) ( ) 3
c c P A B P A B P A B PB A
Adding equation (1) and equation (2) then we get
P( A) P(B) [P(A B ) P(A B) P(B A)] P(A B)
c c
P( A)P(B)P(AB)P(AB) ( fromequation(3))
P( AB)P(A)P(B)P(AB)
Boole’s Inequality : If A 1 , A 2 , A 3 ,.... An be n events such that
(^)
n
i
i
n
i
PA n i
i P A
1 1
() ( ) ( 1 )
^
n
i
ii Ai P Ai 1
n
i 1
( ) P ( )
s
Proof : (i) Let A 1 , A 2 be any two events then by the addition theorem of probability
P( A 1 A 2 )P(A 1 )P(A 2 )P(A 1 A 2 )
By the properties of probability ,
0 P(A 1 A 2 ) 1
0 P(A 1 )P(A 2 )P(A 1 A 2 ) 1
P( A 1 ) P(A 2 )P(A 1 A 2 ) 1
P( A 1 )P(A 2 ) 1 P(A 1 A 2 )
P( A 1 A 2 )P(A 1 )P(A 2 ) 1 ( 1 )
P( A 1 A 2 )P(A 1 )P(A 2 )( 2 1 )
2
1
2
1
(^) i
i i i
P A P A
The given expression is true for n =
Let the given expression is true for n= k
1
1
(^)
P A P A k
k
i
i i
k
i
Consider ( 1 2 3 ......... Ak Ak 1 )
1
1
P Ai P A A A
k
i
1 1 i k
k
i
P A A
( 1 ) 1 [fromequation(2)]
P Ai P Ak
1
k
k
i
P Ai k P A
1
1
1
1
(^)
P P A k
k
i
i
k
i
Put n = k+1 in the above equation, then we get
1 1
(^)
P P A n
n
i
i
n
i
The given expression is true for n=k+
By mathematical induction , the given expression is true for all real values of ‘n’
(ii)
n
i
i i
n
i
P A PA
1 1
Let A 1 , A 2 be any two events then by the addition theorem of probability
P( A 1 A 2 )P(A 1 )P(A 2 )P(A 1 A 2 )
P( A 1 A 2 )P(A 1 )P(A 2 )P(A 1 A 2 )
By the properties of probability
P( A 1 A 2 ) 0
P( A 1 ) P(A 2 )P(A 1 A 2 ) 0
i i i
P Ai P A
2
1
2
1
2
1
2
1 i i
i i
P A P A
The given expression is true for n=
Let the given expression is true for n=k then
1 1
(^) i
k
i
i
k
i
P A P A
Consider
(( 1 2 3 ......... Ak) 1 )
1
1
(^) i k
k
i
P A P A A A A
1 1 i k
k
i
P A A
1 1 1 1
1
1
( ) i k
k
i i k
k
i i
k
i
P A P A PA P A A
i
k
i i k
k
i i k
k
i
P A A P A P A P A
1
1 1 1 1 1
1
1 1 1
i
k
i i k
k
i
P A P A P A
k
i
i
k
i
PAi P Ak P A 1
1
1
RESULT : If A and B are independent then prove that
(i) A
c and B are independent (ii) A and B
c are independent (iii) A
c and B
c are independent
Proof : Given A and B are independent
i.e., P(A B)P(A).P(B) (1)
from the Venn diagram
(i) Consider
B A B ( A B)
c
Taking probability on both sides, then we get
P B P A B A B
c ( )
P A B P A B P A B A B
c c
P B P A B P A B
c ( )
P A B PB P A B
c ( )
P ( B)P(A) P(B) P( B)( 1 P(A)) ( ) ( )
c PB P A
A
c and B are independent.
(ii) Consider A ^ A B ^ ( A B)
c
Taking probability on both sides, then we get
P A P A B A B
c ( )
P A B P A B P A B A B
c c
P A P A B P A B
c ( )
P A B P A P A B
c ( )
P ( A)P(A) P(B) P( A)( 1 P(B))
c P AP B
A and B
c are independent.
(iii) ConsiderP A B 1 P(A B)
c c
1 P ( A)P(B)P(AB) 1 P( A)P(B)P(AB)
1 P (A)P(B)P(A).P(B) 1 [ 1 P (A)]P(B)[ 1 P(A)]
( ).P(B)
[ 1 ( )][ 1 ( )]
c c P A
P A PB
P A ( ) ( )
c c c c B P B P A
A
c and B
c are independent.
UNIT – II
RANDOM VARIABLE
Random variable : Any real valued function which is defined on a sample space is known as a random
variable.
(or)
A variable which takes values from a random experiment is known as a random variable.
Ex : If we toss THREE coins at a time then sample space S = { TTT, TTH, HTH, THH, HTT, THT, HHT, HHH }
If we assumed that “ H “ is a favorable case and the number of heads appeared is taken as a
random variable ‘ x ‘ then it takes the values 0, 1, 2, 3
Types of Random variables : Random variables are classified into TWO types. They are
- Discrete Random variable
- Continuous Random variable 1. Discrete Random Variable : A random variable which takes finite number of values , then such variable
is known as discrete random variable.
EX : In rolling a die , a random variable ‘ x’ takes 6 values. i. e., S= { 1, 2, 3, 4, 5, 6 }
2. Continuous Random variable : A random variable which takes all possible values between certain limits
is known as continuous random variable.
(or)
If a random variable takes infinite number of values then such variable is known as continuous
random variable.
Probability Function : Probability function can be defined in two types. They are
- Probability mass function 2. Probability density function 1. Probability Mass Function : If ‘ x’ is a discrete random variable then the probability function
P( x)P(X x) x is called probability mass function. If and only if it satisfies
P( x) 0
0 P(x) 1
n
x
P x 0
2. Probability Density function : If ‘ x’ is a continuous random variable then the probability function
f(x) is given by x
dx x x
dx f x P x 2 2
is called probability density function and it
must satisfies the following conditions.
f( x) 0
0 f(x) 1
f( x)dx 1
Distribution Function : distribution functions are of TWO types. They are :
- Discrete distribution function
- Continuous distribution function
Properties of Mathematical Expectation :
- Mathematical expectation of a constant ‘k’ is also a constant. i.e., E(k)=k
2. If ‘ x’ is a random variable and ‘k ‘ is a constant then E ( kx)kE(x)
3. If ‘ x’ and ‘ y’ are any two random variables thenE ( xy)E(x)E(y)
4. If ‘ x’ is a random variable and a, b are any two constants thenE (ax b)aE(x)b
5. If x and yare any two random variables and a, b are the constants then
E( axby)aE(x)bE(y)
- If x 1 , x 2 ,x 3 ...... xnbe the ‘n‘ random variables then
E( x 1 x 2 x 3 .......xn )E(x 1 )E(x 2 )E(x 3 )..........E(xn)
7. If ‘ x’ and ‘ y’ are any two independent random variables then E ( xy)E(x) E(y)
- If x 1 , x 2 ,x 3 ...... xnbe the ‘n ‘ mutually independent random variables then
E( x 1. x 2 .x 3 ......xn )E(x 1 )E(x 2 )E(x 3 )........E(xn) ( ) 1 1 i
n
i i
n
i
E x E x
^
- If g(x)is a function of the discrete random variable ‘ x’ then (^)
n
i
E g x g xi P xi 1
- If g(x)is a function of the continuous random variable ‘ x’ then
E g(x) g(x)f(x)dx
Uses of Mathematical expectation :
Mathematical expectation can be used to
- Find the two moments and central moments of the data.
- Find the variance of the given random variable.
- Determine covariance between two random variables.
- To construct sampling distribution of the sample mean.
- Mathematical expectation plays an important roll in probability distribution, sampling techniques ,
large and small sample tests.
Addition theorem of expectation :
Statement : If x and y are any two random variables thenE( xy)E(x)E(y)
Proof : We can prove the addition theorem on expectation in two cases
(i) Discrete case (ii) Continuous case
(i) Discrete case : If x is a discrete random variable with the probability mass function P(x) then it’s
mathematical expectation is given by ( ) ( ) ( 1 ) 1
(^)
n
i
E x xi P xi
If (^) y is a discrete random variable with the probability mass function P( y) then it’s
mathematical expectation is given by ( ) ( ) ( 2 ) 1
(^)
n
j
E y yj P yj
Let x y is a discrete random variable with the probability function P( xi yj)then it’s
mathematical expectation is given by
n
i
i j
n
j
E x y xi yj P x y 1 1
=
n
ii
n
i
n
j
j i j
n
j
xi P xiyj y P xy 1 1 1
=
n
j
n
i
j i j
n
i
i j
n
j
xi Pxy y P xy 1 1 1 1
=
n
j
j j
n
i
xi Pxi y P y 1 1
E(x y)E(x)E(y)
Y 1 Y 2 Y 3.......... Yj..... Yn Total
X 1 P(x 1 y 1 ) P(x 1 y 2 ) P(x 1 y 3 ) (^) ……………… P(x 1 yj ) (^) ……………… P(x 1 yn ) P(x 1 )
X 2 P(x 2 y 1 ) P(x 2 y 2 ) P(x 2 y 3 ) (^) ……………… P(x 2 yj ) (^) ……………… P(x 2 yn ) P(x 2 )
X 3 P(x 3 y 1 ) P(x 3 y 2 ) P(x 3 y 3 ) (^) ……………… P(x 3 yj ) (^) ……………… P(x 3 yn ) P(x 3 )
……………… ……………… ……………… ………………
……………… ………………
……………… ……………… ………………
xi P(xi y 1 ) P(xi y 2 ) P(xi y 3 ) (^) ……………… P(xi yj ) (^) ……………… P(xi yn ) P(xi)
……. ……………… ……………… ………………
……………… ………………
……………… ……………… ………………
Xn P(xn y 1 ) P(xn y 2 ) P(xn y 3 ) (^) ……………… P(xn yj ) (^) ……………… P(xn yn ) P(xn)
Total P(y 1 ) P(y 2 ) P(y 3 ) (^) ……………… P(yj) (^) ……………… P(yn) 1
(ii) Continuous Case : If xis a continuous random variable then it’s mathematical expectation is
given by ( ) ( ) (3) E x x f x dx x
If (^) yis a continuous random variable then it’s mathematical expectation is given by
E y yf y dy
x
Let x yis a continuous random variable then it’s mathematical expectation is given by
x y
E (x y) (x y)f(x,y)dx dy
x y x y
x f(x,y)dxdy y f(x,y)dx dy
= x f x y dydx y f x y dxdy
x y y x
y
f x y dx f y
f(x,y)dy f x
x
x y
x f(x)dx yf(y) dy
E( xy)E(x)E(y)
Variance : If ‘ x’ is a random variable then it’s variance is given by
2 V ( x)ExE(x )
( ) 2 ( )
2 2 E x Ex xEx
( ) ( ) 2 ( ) ( )
2 2 E x E Ex ExEx
2 2 2 E( x ) E(x) 2 E(x)
2 2 E( x )E(x)
If xis a discrete random variable then it’s variance is given by
2 2 V ( x)E(x)E(x)
2
1 1
2 ( ) ( )
(^)
i
n
i
i
n
i
xi Pxi x P x
If xis a continuous random variable then it’s variance is given by
2 2 V ( x)E(x)E(x)
2 2 ( ) ( )
x x
x f x dx xf x dx
Properties of variance :
- Variance of a random variable is always non- negative. i.e.,V( x) 0
- Variance of constant is equal to zero. i.e., V(k)=
3. If xis a random variable and k is a constant then
a. ( ) ( )
2 V kx k V x
b. ( )
2 V x k k
x V (^)
c. V ( xk)V(x)
4. If xis a random variable and a, b are any two constants. Then
a. ( ) ( )
2 V axb aV x
b. ( )
2 V x b b
x a V (^)
5. If xand yare any two random variables then
a. V ( xy)V(x)V(y) 2 cov(xy) If xandyarenotindependent
b. V ( xy)V(x)V(y) If (^) xandyareindependent
6. If xand yare any two random variables and a, b are any two constants then
( ) ( ) ( ) 2 cov( ) If and arenotindependent
2 2 V axby aV x bV y xy x y
( ) ( ) ( ) If and areindependent
2 2 V axby aV x bV y x y
- If x 1 , x 2 ,x 3 ............xnare any ‘n ‘ random variables , a 1 , a 2 ,a 3 ............anare any ‘ n ‘ constants then
^
i j n
i j i j
n
i
i i
n
i
V ai xi aV x aa xy 1 1
2
1
( ) 2 cov
If x 1 , x 2 ,x 3 ............xnare mutually independent random variables then
n
i
i i
n
i
V ai xi aV x 1
2
1
Two dimensional random variables : Let x and y be any two random variables defined on a sample
space ‘ S’ then the function ( (^) x,y) that assigns a probability is called a two dimensional random variable.
Let ( x,y) be a two dimensional random variable defined on a sample space ‘S’ and S then the
value of (^) x,yat is given by the pair of real numbersX ( ),Y( )
Let all the events S such that X ( )a and Y ( )b then the probability of an event is
denoted asP ( xa,yb)
Joint Probability Function : If ( x,y) is a two dimensional discrete random variable then the joint discrete
function of x,yis called As the joint probability mass function of ( x,y) and it is denoted by PXYand it is
defined as PXY x (^) iyj P Xxi,Yyj. If xand yare independent then PXY ( xi,yj) 0
Y 1 Y 2 Y 3.......... Yj..... Yn Total
X 1 P 11 P 12 P 13 ……………… P1j ……………… P1n P.
X 2 P 21 P 22 P 23 ……………… P2j ……………… P2n P.
X 3 P 31 P 32 P 33 ……………… P3j ……………… P3n P.
……………… ……………… ……………… ………………
……………… ………………
……………… ……………… ………………
xi Pi1 Pi2 Pi3 ……………… Pij ……………… Pin Pi.
……. ……………… ……………… ………………
……………… ………………
……………… ……………… ………………
Xn Pn1 Pn2 Pn3 ……………… Pnj ……………… Pnn Pn.
Total P. 1 P. 2 P. 3 ……………… P. j ……………… P. n 1
