**Probability and Statistics Notes:** Probability and statistics are different fields individually as well but are often used in combination for academic and research purposes. Probability talks about favourable outcomes for any event in numerical terms. Statistics is a field that is concerned with the collecting, organizing, analysing, interpretation and representation of data. This data is usually derived from random experiments during research.

Students should have a better understanding of the subject and to help them in their understanding of the subject, they can use Probability and statistics Lecture Notes pdf for JNTU to help them in their preparations of the exam. For further details, click on any one of the links given below:

## Introduction to Probability and Statistics Notes

- Probability And Statistics Notes PDF
- Reference books for Probability and statistics
- Probability and Statistics Syllabus
- Theoretical problems of Probability and statistics
- Frequently asked Questions
- Conclusion

### Probability and Statistics Notes PDF

The pdf is among the few reliable and accurate study materials available online. These are well-categorized notes to increase its ease of use. Also, it has topic wise categorizations so you can easily go back and bolster your weak points if any. Plus it’s great to be used as a revision tool. You can quickly skim through each chapter and brush up your memory of all the major topics so that you can score well in your semester exams.

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### Reference Books for Probability and statistics

Given below is the list of books recommended for the probability and statistics paper. Finding a reliable reference book online is no easy task. With the amount of information available, the guarantee of it being accurate, reliable and up to date is comparatively low. This list of books is all verified by experts and recommend these books themselves. Also, these are recommended by teachers and students alike. Having a reliable source of information will come in handy in your preparations.

Most of the books recommended in this list are some of the only few books students can use for their preparations. This list is in no particular order. It is advisable that if newer editions of these are available, students should opt for them because they will include any update in the syllabus and will also be the most up to date which will help you in your preparations a lot.

- Higher Engineering mathematics, Dr B.S. Grewal ( Khanna publishers)
- Probability and statistics for engineers and scientists, Sheldon M. Ross( Academic Press)
- Operations research, S.D. Sharma
- Mathematics for engineers, K.B. Dutta and M.S.Sriniva (Cengage publications)
- Probability and statistics, TKV Iyengar
- Fundamentals of mathematical statistics, S.C. Gupta
- An introduction to statistical learning: with application in R, Daniela Witten(2013)
- Introduction to probability, statistics and random processes, Hossien Pishro-Nik(2014)

### Probability and Statistics Syllabus

The entire syllabus of the syllabus covers a lot of ground. The field of probability and statistics is ever-growing with newer additions to this field thanks to the success in the research fields. Probability and statistics find its use in many fields, and therefore its applications are many and still counting. Students when preparing for their exams should keep in mind that the entire syllabus is interconnected, so it does require constant effort from their side.

The syllabus is divided into five units. Each unit is a point of discussion in itself. Students should definitely look and make note that they are familiar with the topics and the understanding of the topic as well.

**Unit 1**

Single Random variables and probability distributions

Random variables – Discrete and continuous. Probability distributions, mass function/ density function of a probability distribution. Mathematical Expectation, Moment about the origin, Central moments Moment generating function of a probability distribution. Binomial, Poisson & normal distributions and their properties. Moment generating functions of the above three distributions, and hence finding the mean and variance.

**Unit 2**

Multiple Random variables, Correlation & Regression

Joint probability distributions- Joint probability mass/ density function, Marginal probability mass / density functions. The covariance of two random variables, Correlation Coefficient of correlation, The rank correlation. Regression- Regression Coefficient, The lines of regression and multiple correlation & regression.

**Unit 3**

Sampling Distributions and Testing of Hypothesis

Sampling: Definitions of population, sampling, statistic, parameter. Types of sampling, Expected values of Sample mean and variance, sampling distribution, Standard error, the sampling distribution of mean and the sampling distribution of variance.

Parameter estimation – likelihood estimate, interval estimations.

Testing of hypothesis

Null hypothesis, Alternate hypothesis, type I, & type II errors – critical region, confidence interval, Level of significance, Once sided test, Two-sided test,

Large sample tests:

Test of Equality of means of two samples equality of sample mean and population mean (cases of known variance & unknown variance, equal and unequal variances)

Tests of significance of the difference between sample S.D and population S.D.

Tests of significant difference between sample proportion and population proportion & difference between the two sample proportions.

Small sample tests

Student t-distribution, its properties; Test of significant difference between the sample mean and population mean; the difference between means of two small samples.

Snedecor’s F-distribution and its properties. Test of equality of two population variances.

Chi-square distribution, it’s properties, Chi-square test of goodness of fit.

**Unit 4**

Queuing Theory

Structure of a queuing system, Operating characteristics of queuing system, Transient and steady states, Terminology of Queuing systems, Arrival and service processes- Pure Birth-Death process Deterministic queuing models- M/M/1 Model of infinite queue, M/M/1 model of the finite queue.

**Unit 5**

Stochastic processes

Introduction to Stochastic Processes – Classification of Random processes, Methods of the description of random processes, Stationary and non-stationary random processes, Average values of single random processes and two or more random processes. Markov process, Markov chain, classification of seats – Examples of Markov Chains, Stochastic Matrix.

### Theoretical Problems of Probability and statistics

The objective of the course is to familiarise the students with the important concepts of probability and statistics such as random variable, binomial and Poisson distribution, sampling, estimation, hypothesis, queuing and random processes. Here is a list of some commonly asked theoretical problems of probability and statistics. Students can be asked to answer them in their exam papers and also during interviews. Keep in mind that these are basic entry-level questions to give you a better sense of the subject and what pattern does the examination paper follow-

- What is queuing theory?
- Comment on different types of sampling.
- What is a random variable?
- What is the normal curve?
- Explain stochastic processes.
- What is the use of Venn diagrams?
- Differentiate between independent and dependent events?
- What is conditional probability?
- What are the applications of probability techniques?
- In how many ways ten people can sit in a circle?
- What are an outlier and inlier?
- Explain normal distribution.
- How do we handle missing data to minimise error?
- What is the mean imputation method?
- Explain the difference between a cause and a correlation.

### Frequently Asked Questions on Probability and Statistics Notes

Question 1

What is the theory of probability?

Answer:

Probability is the study of the possibility of any event in a random experiment. We calculate the probability to determine the possibility that an event occurs or not.

Probability is defined as:

P= Number of favourable outcome/Total number of outcomes

Also,

P+P=1

Where,

P=event does not occur.

P= occurrence of an event

Equally likely events – These are events which have a similar probability of occurring. For example, during a coin toss, the probability of heads or tails is equal, i.e., ½.

Complementary events – Events which are opposite of each other. It means that if one of them occurs then the second doesn’t. Example of a complimentary event is will it rain or not tomorrow.

Question 2

Explain Normal distribution.

Answer:

The normal distribution is a distribution that follows a bell curve. In a normal distribution, there is a symmetry about the centre. At the centre point lies the mean, median and mode.

Half the values are above and below this mean value. In nature, a perfect normal distribution may not exist, but some cases tend towards this normal distribution. For example- Heights of people, marks on a test etc. closely resemble the normal distribution.

Also, if the number of values increases, it starts moving towards a normal distribution.

Concepts of the normal distribution are used to find values like standard deviation about the mean and z-core. These are commonly asked in the examination.

Standard deviation, = square root of the variance

Variance, 2=((x1-)2+(x2-)2+….+(xn-)2)/n

Z-score, Z=(x-)/

Question 3

Find the probability that a leap year has 52 Sundays?

Answer:

This is a very basic problem in probability. This is a calendar question and is good to brush up your memory to brush up your concepts of probability.

Solution:

A leap year has 366 days, i.e., one day more than in a normal year. It has a total of 52 weeks and two additional days.

There are 52 Sundays in a leap year. Now the two days can be :

**Case1:** Saturday, Sunday

**Case2:** Sunday, Monday

**Case3:** Monday, Tuesday

**Case4:** Tuesday, Wednesday

**Case5:** Wednesday, Thursday

**Case6:** Thursday, Friday

**Case7:** Friday, Saturday

Now in case one and case two, there is a Sunday. The number of cases without Sundays is 5.

therefore, the probability comes out to be:

P=5/7

Question 4

Explain Selection bias.

Answer:

During the selection of a sample, group, data or individuals, certain biases come into play. These biases are not a clear representation of the population, and therefore, proper randomization can’t be achieved. It is very important to know about these selection biases because they can produce false results regarding a population.

- Sampling bias: Any biases involved in the sample, which is caused due to non-random sampling, will not give a clear representation of the population.
- Time interval- Using a specific period to calculate and support the argument. For example,- Sales analysis during the festive period will not represent the population accurately.
- Data: Any biases made when picking some data samples over others. For example- cherry-picking certain values to prove the argument in question.

Any missing data can also affect the calculations because null value, if added, will make the sample more spread and which might not be the actual case.

### Conclusion

Probability and statistics notes pdf JNTU is a really useful study material for students looking to score well in their exams or improve their concepts of the subject. Probability and statistics are subjects where you need ample theoretical knowledge, and you should be clear about their applications as well. If you have a good demand over mathematics, that will be helpful for you. The calculations are often complex, and students make mistakes in them a lot. It’s suggested that you download the free pdf so that you have a reliable study material you can consult whenever you feel that you are stuck.