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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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