The confidence interval for the expectation of a normal sample

Lecture

The case of known variance [edit]

Let be The confidence interval for the expectation of a normal sample - independent sampling from the normal distribution, where - known dispersion. Define arbitrary and build a confidence interval for the unknown mean .

Statement. Random value

The confidence interval for the expectation of a normal sample

has a standard normal distribution The confidence interval for the expectation of a normal sample . Let be - -quantile standard normal distribution. Then, due to the symmetry of the latter, we have:

The confidence interval for the expectation of a normal sample

.

After substitution of the expression for The confidence interval for the expectation of a normal sample and simple algebraic transformations we get:

The confidence interval for the expectation of a normal sample

.

The case of unknown variance [edit]

Let be The confidence interval for the expectation of a normal sample - independent sampling from the normal distribution, where - unknown constants. Construct a confidence interval for an unknown mean .

Statement. Random value

The confidence interval for the expectation of a normal sample

,

Where The confidence interval for the expectation of a normal sample - unbiased sample standard deviation, has a Student’s distribution with degrees of freedom . Let be - quantile student distribution. Then, due to the symmetry of the latter, we have:

The confidence interval for the expectation of a normal sample

.

After substitution of the expression for The confidence interval for the expectation of a normal sample and simple algebraic transformations we get:

The confidence interval for the expectation of a normal sample

.

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Lectures and tutorial on "Probability theory. Mathematical Statistics and Stochastic Analysis"

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis

Event. Random events. Event probability

Discrete random variables

Continuous random variables

Mathematical statistics

Laplace function and other tabular statistical functions: in Excel formula

Tables The value of the Laplace function f (x), tγ, q Critical points of the distribution of χ2 and Student

Expected value

Random Variance

Correlation

Regression analysis

Coefficient of determination

Linear regression

Least square method

Confidence interval

The confidence interval for the expectation of a normal sample

Confidence interval for normal sample variance

Normal distribution

Generating function

Generating function of moments

Generating function of a sequence (gentris)

Generating function of canonical transformation

Generatrix

Elementary event space

Numerical Characteristics of Random Variables

Poisson formula and an example of solving a problem

Bernoulli formula and an example problem solution

Bayes formula and examples of tasks

Full probability formula

The addition theorem for probabilities of incompatible events

Random events Event algebra Classical and statistical definitions of the probability of an event

the table of the main generating functions and the proof (conclusion).

Uniform distribution of random variable.

Geometric distribution

Binomial distribution

Poisson distribution

Standard normal distribution

Exponential distribution

. Multidimensional random variables. Random functions

Correlation coefficient properties

Distribution density of a system of two random variables

Polygon and histogram

Variance analysis

Random moments

Bertrand Paradox (probability)

Galton's board (quincunks) to demonstrate the central limit theorem

Probability

Probability distribution

Probability density

Quantile

Bayesian Spam Filtering

Rows of distribution. Polygon, Bar Graph, Cumulate, Ogiva

Summary and grouping of statistical data, Concept and types of grouping, Principles of grouping, Secondary grouping

Population and sampling method, Sampling errors, Sampling volume required

Types and analysis of time series. Methods for calculating the average level in the series of dynamics

Correlation and regression analysis. Linear correlation

Blind Signature Blind Signature

Random graph

2.5. Almost impossible and almost reliable events. The principle of practical universality

1.1. Subject of probability theory

1.2. Brief historical information

2.2. Direct calculation of probabilities

2.3. Frequency, or statistical probability, of an event

3.1. Purpose of the main theorems. Sum and Event

3.2. Probability addition theorem

3.3. Probability multiplication theorem

3.4. Full probability formula

3.5. Theorem of hypotheses (Bayes formula)

4.1. Private theorem on repetition of experiments

4.2. General repetition theorem

5.1. A number of distribution. Polygon distribution

5.2. Distribution function

5.3. The probability of hitting a random variable on a given area

5.4. Distribution density

5.5. Numerical characteristics of random variables. Their role and purpose

5.6. Position characteristics (expected value, mode, median)

5.7. Moments. Dispersion. Standard deviation

5.8. Uniform density law

5.9. Poisson law

6.1. Normal distribution law and its parameters

6.2. Moments of normal distribution

6.3. The probability of hitting a random variable in a given area. Normal distribution function

6.4. Probable (median) deviation

7.1. The main tasks of mathematical statistics

7.2. Simple statistical aggregate. Statistical distribution function

7.3. Statistical series. bar chart

7.4 Numerical Characteristics of the Statistical Distribution

7.5. Alignment of statistical series

7.6. Consent Criteria

8.1. The concept of a random variable system

8.2. The distribution function of the system of two random variables

8.3. Distribution density of a system of two random variables

8.4. The laws of distribution of individual quantities included in the system. Conditional laws of distribution

8.5 Dependent and independent random variables

8.6. Numerical characteristics of a system of two random variables. Correlation moment. Correlation coefficient

8.7. System of an arbitrary number of random variables

8.8. Numerical characteristics of a system of several random variables

9.1. Normal law on the plane

9.2 Ellipses of dispersion. Reduction of the normal law to canonical form

9.3. The probability of hitting a rectangle with sides parallel to the main axes of dispersion

9.4. The probability of hitting an ellipse

9.5. Probability of hitting an area of arbitrary shape

9.6. Normal law in the space of three dimensions.

10.1. Expectation function. Function dispersion

10.2. Numerical Characteristics Theorems

10.3. Applications of theorems on numerical characteristics

11.1. Method of linearization of functions of random arguments

11.2. Linearize the function of one random argument

11.3. Linearization of the function of several random arguments

11.4. Refinement of the results obtained by the method of linearization

12.1. The distribution law of the monotone function of one random argument

12.2. The law of distribution of a linear function from an argument subject to the normal law

12.3. The distribution law of a nonmonotonic function of one random argument

12.4. The distribution law of the function of two random variables

12.5. The distribution law of the sum of two random variables. The composition of the laws of distribution

12.6. Composition of normal laws

12.7. Linear functions of normally distributed arguments

12.8. Composition of normal laws on the plane

13.1. The law of large numbers and the central limit theorem

13.2. Chebyshev Inequality

13.3. The law of large numbers (Chebyshev theorem)

13.4. The generalized Chebyshev theorem. Markov's theorem

13.5. Consequences of the law of large numbers: theorems of Bernoulli and Poisson

13.6. Mass random phenomena and the central limit theorem

13.7. Characteristic functions

13.8. Central limit theorem for equally distributed terms

13.9. Formulas expressing the central limit theorem and occurring in its practical application

14.1. Features of processing a limited number of experiments. Estimates of the length of the unknown parameters of the distribution law

14.2. Estimates for expectation and variance

14.3. Confidence interval Confidence probability

14.4. Exact methods for constructing confidence intervals for random variable parameters

14.5. Frequency probability estimate

14.6. Estimates for the numerical characteristics of a system of random variables

14.7. the task of processing the results of experimental firing (bombing).

14.8. Smoothing of experimental dependencies by the least squares method

15.1. Theory of random functions. Concept of random function

15.2. The concept of a random function as an extension of the concept of a system of random variables. Distribution law of a random function

15.3. Characteristics of random functions

15.4. Characterization of a random function from experience

15.5. Methods for determining the characteristics of the transformed random functions from the characteristics of the original random functions

15.6. Linear and nonlinear operators. Dynamic system operator

15.7. Linear transformations of random functions

16.1. The idea of the method of canonical decompositions. Representation of a random function as a sum of elementary random functions

16.2. Canonical decomposition of a random function

16.3. Linear transformations of random functions defined by canonical decompositions

17.1. The concept of stationary random process

17.2. Spectral decomposition of a stationary random function on a finite time interval. Dispersion spectrum

17.3. Spectral decomposition of a stationary random function on an infinite segment of time. Spectral density of stationary random function

17.4. Spectral decomposition of a random function in a complex form

17.5. Transformation of stationary random function by stationary linear system

17.6. Applications of the theory of stationary random processes to solving problems associated with the analysis and synthesis of dynamic systems

17.7. Ergodic property of stationary random functions

17.8. Characterization of an ergodic stationary random function in one implementation

The Simpson paradox in statistics with examples