Electrical Engineering (EE) Exam > Electrical Engineering (EE) Notes > Short Notes for Electrical Engineering > Short Notes: Probability & Statistics
Short Notes: Probability & Statistics | Short Notes for Electrical Engineering - Electrical Engineering (EE) PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
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
Read More
About this Document
|
4.70/5 Rating |
|
Dec 22, 2024 Last updated |
Document Description: Short Notes: Probability & Statistics for Electrical Engineering (EE) 2024 is part of Short Notes for Electrical Engineering preparation.
The notes and questions for Short Notes: Probability & Statistics have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about Short Notes: Probability & Statistics covers topics
like and Short Notes: Probability & Statistics Example, for Electrical Engineering (EE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Short Notes: Probability & Statistics.
Introduction of Short Notes: Probability & Statistics in English is available as part of our Short Notes for Electrical Engineering
for Electrical Engineering (EE) & Short Notes: Probability & Statistics in Hindi for Short Notes for Electrical Engineering course.
Download more important topics related with notes, lectures and mock test series for Electrical Engineering (EE)
Exam by signing up for free. Electrical Engineering (EE): Short Notes: Probability & Statistics | Short Notes for Electrical Engineering - Electrical Engineering (EE)
Description
Full syllabus notes, lecture & questions for Short Notes: Probability & Statistics | Short Notes for Electrical Engineering - Electrical Engineering (EE) - Electrical Engineering (EE) | Plus excerises question with solution to help you revise complete syllabus for Short Notes for Electrical Engineering | Best notes, free PDF download
Information about Short Notes: Probability & Statistics
In this doc you can find the meaning of Short Notes: Probability & Statistics defined & explained in the simplest way possible. Besides explaining types of
Short Notes: Probability & Statistics theory, EduRev gives you an ample number of questions to practice Short Notes: Probability & Statistics tests, examples and also practice Electrical Engineering (EE)
tests
|
Explore Courses for Electrical Engineering (EE) exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Important questions
,ppt
,MCQs
,Objective type Questions
,Summary
,Short Notes: Probability & Statistics | Short Notes for Electrical Engineering - Electrical Engineering (EE)
,Sample Paper
,mock tests for examination
,shortcuts and tricks
,Previous Year Questions with Solutions
,practice quizzes
,Semester Notes
,Extra Questions
,study material
,Short Notes: Probability & Statistics | Short Notes for Electrical Engineering - Electrical Engineering (EE)
,Exam
,video lectures
,past year papers
,Viva Questions
,Free
,Short Notes: Probability & Statistics | Short Notes for Electrical Engineering - Electrical Engineering (EE)
;
Additional Information about Short Notes: Probability & Statistics for Electrical Engineering (EE) Preparation
Short Notes: Probability & Statistics Free PDF Download
The Short Notes: Probability & Statistics is an invaluable resource that delves deep into the core of the Electrical Engineering (EE) exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Short Notes: Probability & Statistics now and kickstart your journey towards success in the Electrical Engineering (EE) exam.
Importance of Short Notes: Probability & Statistics
The importance of Short Notes: Probability & Statistics cannot be overstated, especially for Electrical Engineering (EE) aspirants.
This document holds the key to success in the Electrical Engineering (EE) exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Short Notes: Probability & Statistics
Short Notes: Probability & Statistics Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Short Notes: Probability & Statistics.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Short Notes: Probability & Statistics Notes on EduRev are your ultimate resource for success.
Short Notes: Probability & Statistics Electrical Engineering (EE) Questions
The "Short Notes: Probability & Statistics Electrical Engineering (EE) Questions" guide is a valuable resource for all aspiring students preparing for the
Electrical Engineering (EE) exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Short Notes: Probability & Statistics on the App
Students of Electrical Engineering (EE) can study Short Notes: Probability & Statistics alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Short Notes: Probability & Statistics,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Short Notes: Probability & Statistics is prepared as per the latest Electrical Engineering (EE) syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup