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 Page 1


 
 
 
 
 
PROBABILITY AND STATISTICS 
 
Types of events 
? Complementary events  
  
? ? ? ? ? ?
c
E s E ??  
  The complement of an event E is set of all outcomes not in E.  
 
? Mutually Exclusive Events  
  Two events E & F are mutually exclusive iff P(E n F) = 0.  
 
? Collectively exhaustive events 
  Two events E & F are collectively exhaustive iff (E U F) = S   
  Where S is sample space.  
 
? Independent events  
  If E & F are two independent events  
   P(E n F) = P (E) * P(F)  
 
De Morgan’s Law  
?  
? ?
??
??
??
C
C
i
n n
i
i1 i1
 E =  E
U
     
?  
??
??
??
??
??
C
C
i
nn
i
i 1 i 1
 E =  E    
 
Axioms of Probability  
 
n
12
E ,E ,...........,E  are possible events & S is the sample space.  
a. 0 = P (E) = 1  
 
b. P(S) = 1 
  
c. ? ?
n n
ii
i=1 i1
P E = P E
?
??
??
??
?
      for mutually exclusive events  
 
 
 
Page 2


 
 
 
 
 
PROBABILITY AND STATISTICS 
 
Types of events 
? Complementary events  
  
? ? ? ? ? ?
c
E s E ??  
  The complement of an event E is set of all outcomes not in E.  
 
? Mutually Exclusive Events  
  Two events E & F are mutually exclusive iff P(E n F) = 0.  
 
? Collectively exhaustive events 
  Two events E & F are collectively exhaustive iff (E U F) = S   
  Where S is sample space.  
 
? Independent events  
  If E & F are two independent events  
   P(E n F) = P (E) * P(F)  
 
De Morgan’s Law  
?  
? ?
??
??
??
C
C
i
n n
i
i1 i1
 E =  E
U
     
?  
??
??
??
??
??
C
C
i
nn
i
i 1 i 1
 E =  E    
 
Axioms of Probability  
 
n
12
E ,E ,...........,E  are possible events & S is the sample space.  
a. 0 = P (E) = 1  
 
b. P(S) = 1 
  
c. ? ?
n n
ii
i=1 i1
P E = P E
?
??
??
??
?
      for mutually exclusive events  
 
 
 
 
 
 
 
 
Some important rules of probability 
 P(A U B) = P(A) + P(B) – P(A ?B)  
 P(A ? B) = P(A)* P
? ?
B | A = P(B) * P ? ? A | B   
    P ? ? A | B is conditional probability of A given B.  
 If A & B are independent events  
   P(A ?B)  = P(A) * P(B)  
   P(A | B) = P(A)  
   P(B | A) = P(B)    
 
Total Probability Theorem   
  P(A ?B) = P (A ?E) + P (B ?E)  
       = P(A) * P(E |A) + P(B) * P(E |B)    
 
Baye’s Theorem  
   P(A |E) =     P(A ? E) + P (B ?E)         
            =  P(A)* P(E | A) + P(B) * P(E | B)    
 
Statistics  
? Arithmetic Mean of Raw Data   
  
x
x
n
?
?
   
  x = arithmetic mean; x = value of observation ; n = number of observations  
? Arithmetic Mean of grouped data  
    
? ?
?
?
?
fx
x
f
  ;   f = frequency of each observation  
 
? Median of Raw data  
 Arrange all the observations in ascending order  
     
n
12
x x ............ x ? ? ?   
If n is odd, median = 
? ? n1
2
?
 th value  
If n is even, Median =  
? ? ? ?
th th
nn
 value + 1 value
22
2
?
    
 
 
Page 3


 
 
 
 
 
PROBABILITY AND STATISTICS 
 
Types of events 
? Complementary events  
  
? ? ? ? ? ?
c
E s E ??  
  The complement of an event E is set of all outcomes not in E.  
 
? Mutually Exclusive Events  
  Two events E & F are mutually exclusive iff P(E n F) = 0.  
 
? Collectively exhaustive events 
  Two events E & F are collectively exhaustive iff (E U F) = S   
  Where S is sample space.  
 
? Independent events  
  If E & F are two independent events  
   P(E n F) = P (E) * P(F)  
 
De Morgan’s Law  
?  
? ?
??
??
??
C
C
i
n n
i
i1 i1
 E =  E
U
     
?  
??
??
??
??
??
C
C
i
nn
i
i 1 i 1
 E =  E    
 
Axioms of Probability  
 
n
12
E ,E ,...........,E  are possible events & S is the sample space.  
a. 0 = P (E) = 1  
 
b. P(S) = 1 
  
c. ? ?
n n
ii
i=1 i1
P E = P E
?
??
??
??
?
      for mutually exclusive events  
 
 
 
 
 
 
 
 
Some important rules of probability 
 P(A U B) = P(A) + P(B) – P(A ?B)  
 P(A ? B) = P(A)* P
? ?
B | A = P(B) * P ? ? A | B   
    P ? ? A | B is conditional probability of A given B.  
 If A & B are independent events  
   P(A ?B)  = P(A) * P(B)  
   P(A | B) = P(A)  
   P(B | A) = P(B)    
 
Total Probability Theorem   
  P(A ?B) = P (A ?E) + P (B ?E)  
       = P(A) * P(E |A) + P(B) * P(E |B)    
 
Baye’s Theorem  
   P(A |E) =     P(A ? E) + P (B ?E)         
            =  P(A)* P(E | A) + P(B) * P(E | B)    
 
Statistics  
? Arithmetic Mean of Raw Data   
  
x
x
n
?
?
   
  x = arithmetic mean; x = value of observation ; n = number of observations  
? Arithmetic Mean of grouped data  
    
? ?
?
?
?
fx
x
f
  ;   f = frequency of each observation  
 
? Median of Raw data  
 Arrange all the observations in ascending order  
     
n
12
x x ............ x ? ? ?   
If n is odd, median = 
? ? n1
2
?
 th value  
If n is even, Median =  
? ? ? ?
th th
nn
 value + 1 value
22
2
?
    
 
 
 
 
 
 
 
? Mode of Raw data  
Most frequently occurring observation in the data.  
 
? Standard Deviation of Raw Data    
     
? ?
?
??
??
ii
2
2
2
n x x
n
   
    n = number of observations  
   variance = 
2
?   
 
? Standard deviation of grouped data   
    
? ?
?
??
??
2
i i i i
2
2
N f x f x
N
   
    fi = frequency of each observation   
   N = number of observations.  
    variance = 
2
?    
? Coefficient of variation = CV = 
?
?
  
  
?  Properties of discrete distributions  
? ?
?
?
a. P x 1  
? ? ? ?
?
?
b. E X x P x  
? ? ? ? ? ? ? ?
??
2
2
c. V x E x E x   
 
? Properties of continuous distributions  
?  
? ?
?
??
?
?
f x dx 1   
?  
? ? ? ?
??
?
?
x
F x f x dx   =   cumulative distribution  
?  
? ? ? ?
??
?
?
?
E x xf x dx     =  expected value of x   
?  
? ? ? ? ? ?
??
??
??
2
2
V x E x E x  =  variance of x  
 
 
Page 4


 
 
 
 
 
PROBABILITY AND STATISTICS 
 
Types of events 
? Complementary events  
  
? ? ? ? ? ?
c
E s E ??  
  The complement of an event E is set of all outcomes not in E.  
 
? Mutually Exclusive Events  
  Two events E & F are mutually exclusive iff P(E n F) = 0.  
 
? Collectively exhaustive events 
  Two events E & F are collectively exhaustive iff (E U F) = S   
  Where S is sample space.  
 
? Independent events  
  If E & F are two independent events  
   P(E n F) = P (E) * P(F)  
 
De Morgan’s Law  
?  
? ?
??
??
??
C
C
i
n n
i
i1 i1
 E =  E
U
     
?  
??
??
??
??
??
C
C
i
nn
i
i 1 i 1
 E =  E    
 
Axioms of Probability  
 
n
12
E ,E ,...........,E  are possible events & S is the sample space.  
a. 0 = P (E) = 1  
 
b. P(S) = 1 
  
c. ? ?
n n
ii
i=1 i1
P E = P E
?
??
??
??
?
      for mutually exclusive events  
 
 
 
 
 
 
 
 
Some important rules of probability 
 P(A U B) = P(A) + P(B) – P(A ?B)  
 P(A ? B) = P(A)* P
? ?
B | A = P(B) * P ? ? A | B   
    P ? ? A | B is conditional probability of A given B.  
 If A & B are independent events  
   P(A ?B)  = P(A) * P(B)  
   P(A | B) = P(A)  
   P(B | A) = P(B)    
 
Total Probability Theorem   
  P(A ?B) = P (A ?E) + P (B ?E)  
       = P(A) * P(E |A) + P(B) * P(E |B)    
 
Baye’s Theorem  
   P(A |E) =     P(A ? E) + P (B ?E)         
            =  P(A)* P(E | A) + P(B) * P(E | B)    
 
Statistics  
? Arithmetic Mean of Raw Data   
  
x
x
n
?
?
   
  x = arithmetic mean; x = value of observation ; n = number of observations  
? Arithmetic Mean of grouped data  
    
? ?
?
?
?
fx
x
f
  ;   f = frequency of each observation  
 
? Median of Raw data  
 Arrange all the observations in ascending order  
     
n
12
x x ............ x ? ? ?   
If n is odd, median = 
? ? n1
2
?
 th value  
If n is even, Median =  
? ? ? ?
th th
nn
 value + 1 value
22
2
?
    
 
 
 
 
 
 
 
? Mode of Raw data  
Most frequently occurring observation in the data.  
 
? Standard Deviation of Raw Data    
     
? ?
?
??
??
ii
2
2
2
n x x
n
   
    n = number of observations  
   variance = 
2
?   
 
? Standard deviation of grouped data   
    
? ?
?
??
??
2
i i i i
2
2
N f x f x
N
   
    fi = frequency of each observation   
   N = number of observations.  
    variance = 
2
?    
? Coefficient of variation = CV = 
?
?
  
  
?  Properties of discrete distributions  
? ?
?
?
a. P x 1  
? ? ? ?
?
?
b. E X x P x  
? ? ? ? ? ? ? ?
??
2
2
c. V x E x E x   
 
? Properties of continuous distributions  
?  
? ?
?
??
?
?
f x dx 1   
?  
? ? ? ?
??
?
?
x
F x f x dx   =   cumulative distribution  
?  
? ? ? ?
??
?
?
?
E x xf x dx     =  expected value of x   
?  
? ? ? ? ? ?
??
??
??
2
2
V x E x E x  =  variance of x  
 
 
 
 
 
 
 
? Properties Expectation & Variance  
   E(ax + b) = a E(x) + b 
   V(ax + b) = a
2
 V(x)  
   
? ? ? ? ? ?
? ? ?
1 2 1 2
E ax bx aE x bE x  
   
? ? ? ? ? ?
22
1 2 1 2
V ax bx a V x b V x ? ? ?   
   cov (x, y) = E (x y) – E (x) E (y)  
 
Binomial Distribution  
  no of trials = n  
  Probability of success = P  
  Probability of failure = (1 – P)  
      ? ? ? ?
nx
nx
x
P X x C P 1 P
?
? ? ?    
 Mean = E(X) = nP  
 Variance = V[x] = nP(1 – P)  
 
 
Poisson Distribution  
  A random variable x, having possible values 0,1, 2, 3,……., is poisson variable if  
    ? ?
x
e
P X x
x!
??
?
??   
  Mean = E(x) = ?  
  Variance = V(x) = ?    
 
Continuous Distributions 
 
Uniform Distribution  
    
? ?
?
?? ?
?
? ?
?
?
1
 if a x b
fx
ba
0    otherwise
 
  Mean = E(x) = 
ba
2
?
   
  Variance = V(x) = 
? ?
2
ba
12
?
   
 
 
 
Page 5


 
 
 
 
 
PROBABILITY AND STATISTICS 
 
Types of events 
? Complementary events  
  
? ? ? ? ? ?
c
E s E ??  
  The complement of an event E is set of all outcomes not in E.  
 
? Mutually Exclusive Events  
  Two events E & F are mutually exclusive iff P(E n F) = 0.  
 
? Collectively exhaustive events 
  Two events E & F are collectively exhaustive iff (E U F) = S   
  Where S is sample space.  
 
? Independent events  
  If E & F are two independent events  
   P(E n F) = P (E) * P(F)  
 
De Morgan’s Law  
?  
? ?
??
??
??
C
C
i
n n
i
i1 i1
 E =  E
U
     
?  
??
??
??
??
??
C
C
i
nn
i
i 1 i 1
 E =  E    
 
Axioms of Probability  
 
n
12
E ,E ,...........,E  are possible events & S is the sample space.  
a. 0 = P (E) = 1  
 
b. P(S) = 1 
  
c. ? ?
n n
ii
i=1 i1
P E = P E
?
??
??
??
?
      for mutually exclusive events  
 
 
 
 
 
 
 
 
Some important rules of probability 
 P(A U B) = P(A) + P(B) – P(A ?B)  
 P(A ? B) = P(A)* P
? ?
B | A = P(B) * P ? ? A | B   
    P ? ? A | B is conditional probability of A given B.  
 If A & B are independent events  
   P(A ?B)  = P(A) * P(B)  
   P(A | B) = P(A)  
   P(B | A) = P(B)    
 
Total Probability Theorem   
  P(A ?B) = P (A ?E) + P (B ?E)  
       = P(A) * P(E |A) + P(B) * P(E |B)    
 
Baye’s Theorem  
   P(A |E) =     P(A ? E) + P (B ?E)         
            =  P(A)* P(E | A) + P(B) * P(E | B)    
 
Statistics  
? Arithmetic Mean of Raw Data   
  
x
x
n
?
?
   
  x = arithmetic mean; x = value of observation ; n = number of observations  
? Arithmetic Mean of grouped data  
    
? ?
?
?
?
fx
x
f
  ;   f = frequency of each observation  
 
? Median of Raw data  
 Arrange all the observations in ascending order  
     
n
12
x x ............ x ? ? ?   
If n is odd, median = 
? ? n1
2
?
 th value  
If n is even, Median =  
? ? ? ?
th th
nn
 value + 1 value
22
2
?
    
 
 
 
 
 
 
 
? Mode of Raw data  
Most frequently occurring observation in the data.  
 
? Standard Deviation of Raw Data    
     
? ?
?
??
??
ii
2
2
2
n x x
n
   
    n = number of observations  
   variance = 
2
?   
 
? Standard deviation of grouped data   
    
? ?
?
??
??
2
i i i i
2
2
N f x f x
N
   
    fi = frequency of each observation   
   N = number of observations.  
    variance = 
2
?    
? Coefficient of variation = CV = 
?
?
  
  
?  Properties of discrete distributions  
? ?
?
?
a. P x 1  
? ? ? ?
?
?
b. E X x P x  
? ? ? ? ? ? ? ?
??
2
2
c. V x E x E x   
 
? Properties of continuous distributions  
?  
? ?
?
??
?
?
f x dx 1   
?  
? ? ? ?
??
?
?
x
F x f x dx   =   cumulative distribution  
?  
? ? ? ?
??
?
?
?
E x xf x dx     =  expected value of x   
?  
? ? ? ? ? ?
??
??
??
2
2
V x E x E x  =  variance of x  
 
 
 
 
 
 
 
? Properties Expectation & Variance  
   E(ax + b) = a E(x) + b 
   V(ax + b) = a
2
 V(x)  
   
? ? ? ? ? ?
? ? ?
1 2 1 2
E ax bx aE x bE x  
   
? ? ? ? ? ?
22
1 2 1 2
V ax bx a V x b V x ? ? ?   
   cov (x, y) = E (x y) – E (x) E (y)  
 
Binomial Distribution  
  no of trials = n  
  Probability of success = P  
  Probability of failure = (1 – P)  
      ? ? ? ?
nx
nx
x
P X x C P 1 P
?
? ? ?    
 Mean = E(X) = nP  
 Variance = V[x] = nP(1 – P)  
 
 
Poisson Distribution  
  A random variable x, having possible values 0,1, 2, 3,……., is poisson variable if  
    ? ?
x
e
P X x
x!
??
?
??   
  Mean = E(x) = ?  
  Variance = V(x) = ?    
 
Continuous Distributions 
 
Uniform Distribution  
    
? ?
?
?? ?
?
? ?
?
?
1
 if a x b
fx
ba
0    otherwise
 
  Mean = E(x) = 
ba
2
?
   
  Variance = V(x) = 
? ?
2
ba
12
?
   
 
 
 
 
 
 
 
 
 Exponential Distribution  
  ? ?
x
e if x 0
fx
0    if x 0
??
?
? ?
?
?
?
?
?
    
 Mean = E(x) = 
1
?
    
 Variance = V(x) = 
2
1
?
    
 
Normal Distribution  
  ? ?
? ?
2
2
2
x
1
f x e p , x
2
2
??
? ? ?
??
? ? ? ? ? ? ?
??
?
??
??
    
  Means = E(x) = µ    
  Variance = v(x) = 
2
?   
 
Coefficient of correlation  
  
? ?
? ? ? ?
??
cov x, y
var x var y
    
  x & y are linearly related, if ? = ± 1  
  x & y are un-correlated  if ? = 0  
 
Regression lines  
? 
? ?
xx ?  = 
? ?
?
xy
b y y   
? 
? ?
yy ? = 
? ?
?
yx
b x x  
     Where  x & y are mean values of x & y respectively  
 
xy
b = 
? ?
? ?
cov x, y
var y
   ;      
yx
b = 
? ?
? ?
cov x, y
var x
   
 ??
xy yx bb
 
 
 
 
 
 
 
 
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