PROBABILITY AND STATISTICS ENGINEERING MATHEMATICS GATE 2020 STUDY MATERIAL PDF CIVILENGGFORALL

Probability
Sampling
Parameters
Statistics
Standard Error
Population
Conditional Probability
Bayes Rule
Discrete Random Variables
Elements of Probability
Dependent Events
Law of Total Probability
Measures of Central Tendency
Measures of Dispersion
Random Variable
Probability Distribution
Bernoulli Trials
Central Unit Theorem
Gaussian Process
Correlation
Line of Regression
Hypothesis Testing
The Decision Problem

PROBABILITY : It is a concept of mathematics which measures the degree of certainty or uncertainty of the occurrence of events.

If any event can happen in m ways and fails in n ways and each of the (m + n) ways are equally likely to occur, then probability of the happening of the events is defined as the ratio, m/m+n and that of its failing as n/m+n. If probability of the happening is denoted by p and not happening by q, then p + q = 1.
If event is certain to happen, its probability is unity.
If happening is impossible, then its probability is zero.

Sampling : A small section selected from the population is called a sample and the process of drawing a sample is called sampling.

Random Sampling : It is essential that the sample must be a random selection so that each member of the population has the same chance of being included in the sample. Thus the fundamental assumption underlying theory of sampling is random sampling

Simple Sampling : A special case of random sampling in which each event has the same probability p of success and the chance of success of different events are independent whether previous trials have been made or not, is called simple sampling.

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Parameters : The statistical constants of the population such as mean (µ), standard deviation (σ) etc. are called parameters.

Statistics : Constants for the sample drawn from the given population, i.e. mean (x), standard deviation (S) etc. are called statistic. The population parameters are in general, not known and their estimates given by the corresponding sample statistic are used. Greek letters are used to denoted population parameters and Roman letters for sample statistic

Objectives of Sampling

Sampling aims at gathering the maximum information about the population with the minimum effort, cost and time. The object of sampling studies is to obtain the best possible value of the parameters under specific conditions. Sampling determines reliability of these estimates. The logic of the sampling theory is the logic of induction in which we pass from a particular (sample) to general (population). Such a generalization from sample to population is called statistical inference.

Sampling Distribution

Consider all possible sample of size n which can be drawn from a given population at random. For each sample, mean can be computed. The mean of the samples will not be identical. If these different means are grouped according of their frequencies, the frequency distribution so formed is called sampling distribution of the mean. Similarly sampling distribution of the standard deviation etc, can be obtained. When drawing each sample, the previous sample is put back so that the parent population remains the same. This is called sampling with replacement and all subsequent formulae will pertain to sampling with replacements.

Standard Error

It is used to assess the difference between expected and observed values. Standard deviation of the sampling distribution is called standard error (S.E.) Thus standard error of the sampling distribution of means is called standard error of means. The reciprocal of the standard error is called precision. If n≥30, a sample is called large otherwise small. The sampling distribution of large samples is assumed to be normal.

Population

Collection of all possible samples is population. It is impractical to collect data on each sample of a population. Statistics helps to determine best estimate of the population parameter from randomly selected samples. Because random errors are involved in determination of parameters, the estimate will represent a parameter with a given probability only. Given an event B such that P(B) > 0 and any other event A, we define conditional probability of A given by P(A|B) = P(A∩B)/P(B)

ELEMENTS OF PROBABILITY

1. Experiment : It is a process in which a certain work is repeated under the same conditions, the outcomes (or results) of which need not be the same. e.g. tossing a coin or rolling a die are experiments

2. Sample Space: The set S of all possible outcomes of a given experiment is called sample space for the experiment. An outcome an element of S, is called a sample point. e.g. for experiment of tossing a fair (or unbiased) coin sample space S = (H, T) where H, T refer to head, tail respectively.

3. Event: A subset of the sample space S is called an event. The set (a) consisting of a single sample point a in S is called elementary event. Since empty set φ and S are always subsets of S, they are also events φ is called impossible event or null event, and S is called sure or certain event. Given two events A, B. form New events can be for med using the set operations of union, intersection and complementation.

A ꓴ B is the event that occurs if and only if A occurs or B occurs (or both).
A ꓴ B is the event that occurs if and only if both A and B occurs.
A^c, the complement of A, is the event that occurs if and only if A does not occur.