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## Probability and Statistics for Engineers from Imperial College London

### Learn about Probability and Statistics for Engineers. Module given at Imperial College London to second year undergraduate students in electrical and electronic engineering.

Probability and Statistics for Engineers: sets, events, axioms of probability, random variables, probability mass function, probability density function, cumulative distribution function, expectation and variance, single discrete or continuous random variable, systems and component reliability, multiple discrete or continuous random variables, jointly distributed random variables, change of variables, moments, moment generating function, conditional probability, conditional expectation, conditional variance, law of large numbers, central limit theorem, basic properties of estimators, method of moments, and maximum likelihood estimators.

1. Probability and Statistics for Engineers (Part 1 of 8): set theory, events, axioms of probability
2. Probability and Statistics for Engineers (Part 2 of 8): Bayes Theorem, discrete random variables
3. Probability and Statistics for Engineers (Part 3 of 8): discrete and continuous random variables
4. Probability and Statistics for Engineers (Part 4 of 8): continuous random variables
5. Probability and Statistics for Engineers (Part 5 of 8):  jointly distributed random variables
6. Probability and Statistics for Engineers (Part 6 of 8): jointly distributed random variables
7. Probability and Statistics for Engineers (Part 7 of 8): change of variables, LLN, CLT, estimators
8. Probability and Statistics for Engineers (Part 8 of 8): method of moments, maximum likelihood
9. Probability and Statistics for Engineers - Exam Questions

## Probability and Statistics for Engineers (Part 1 of 8): set theory, events, axioms of probability

0:00 Introduction
5:07 what is probability? What is statistics?
10:15 Sets
15:58 Union of sets
19:11 Intersection of sets
24:44 Disjoint sets
26:54 Partition
29:16 Complement of set
32:06 Difference of sets
35:53 Disjoint union
42:14 De Morgan's law
54:36 Sample space and events
1:04:08 Axioms of probability
1:09:28 Probability of union

## Probability and Statistics for Engineers (Part 2 of 8): Bayes Theorem, discrete random variables

Axioms of probability, conditional probability, independence, partition, total probability, Bayes Theorem, discrete random variables, probability mass function (PMF), cumulative distribution function (CDF), expectation, variance.

2:20 Summary of previous lecture
7:30 Conditional probability
12:42 Multiplication law
14:47 two independent events
18:25 mutually independent events
32:38 Probability tables
39:00 Law of total probability
53:00 Bayes theorem
1:01:38 Discrete random variable
1:04:55 Probability mass function (PMF)
1:10:38 Cumulative distribution function (CDF)
1:16:00 Expectation / theoretical mean
1:26:12 Variance and standard deviation

## Probability and Statistics for Engineers (Part 3 of 8): discrete and continuous random variables

Discrete and continuous random variables, probability distribution (Uniform, Binomial, Geometric, Poisson), probability mass function (PMF), cumulative distribution function (CDF), expectation, variance, probability density function (PDF).

0:47 Summary of previous lecture
11:10 Uniform distribution
16:54 Binomial distribution
42:52 Geometric distribution
47:35 Poisson distribution
1:07:17 Continuous random variable
1:09:10 Probability density function (PDF)
1:15:22 Cumulative distribution function (CDF)
1:23:50 Expectation
1:27:25 Variance and standard deviation

## Probability and Statistics for Engineers (Part 4 of 8): continuous random variables

Continuous random variables, probability distribution (Uniform, Exponential, Normal/Gaussian, Chi-Square, Log-Normal), change of variable: one function of one random variable, Chebyshev's Inequality, systems and component reliability.

0:41 Summary of previous lecture
2:53 Uniform distribution
7:17 Exponential distribution
23:29 Change of variable (one to one)
34:30 Normal / Gaussian distribution
1:12:12 Chi square distribution
1:20:22 Log normal distribution
1:26:00 Chebyshev's inequality
1:37:25 Systems and component reliability

## Probability and Statistics for Engineers (Part 5 of 8):  jointly distributed random variables

Systems and component reliability, reliability function, hazard rate, Weibull distribution, jointly distributed random variables, Joint PMF, Joint PDF, marginals, independent random variables, conditional PMF/PDF, conditional expectation.

0:36 Summary of previous lecture
9:30 random time to failure
10:00 Failure time distribution
11:00 Failure time density
11:30 Reliability function.
12:22 Hazard rate
16:10 Cumulative hazard function
26:32 Mean time to failure
28:30 Jointly distributed random variables
30:25 Joint cumulative distribution function
32:30 Joint probability mass function
36:23 Marginal probability mass function
40:30 Joint probability density function
46:50 Marginal probability density function
1:00:05 Independent random variables
1:08:32 Conditional PMF/PDF
1:15:50 Expectation
1:28:36 Conditional expectation

## Probability and Statistics for Engineers (Part 6 of 8): jointly distributed random variables

Conditional variance, covariance, correlation, joint Normal (Gaussian) distribution, moments, moment generating function, characteristic function, relationship with Fourier Transform, sums of random variables.

0:57 Summary of previous lecture
4:52 Conditional variance
18:46 Covariance
33:25 Uncorrelated random variables
43:35 Correlation
51:50 Joint Normal / Gaussian distribution
59:30 Moments
1:03:35 Moment generating function
1:32:45 Characteristic function
1:37:30 Sums of random variables

## Probability and Statistics for Engineers (Part 7 of 8): change of variables, LLN, CLT, estimators

Change of variables: one function of two random variables, two functions of two random variables, Rayleigh distribution, Law of Large Numbers (LLN), Central Limit Theorem (CLT), Statistics and sampling distributions, estimator, properties of estimators, biased/unbiased estimator, minimum variance.

0:00 Summary of previous lecture
5:50 Binomial and Bernoulli
14:46 Change of variables: two to one
16:10 PDF of the maximum of two random variables
21:00 PDF of the sum of two random variables
33:34 Change of variables: two to two
44:04 Rayleigh distribution
54:20 Law of large number and central limit theorem
1:12:10 Statistics
1:16:20 Random sample
1:21:50 Estimator
1:24:50 Unbiased vs biased estimator
1:38:57 Minimum variance unbiased estimator

## Probability and Statistics for Engineers (Part 8 of 8): method of moments, maximum likelihood

Statistics, estimators, mean squared error, method of moments, maximum likelihood estimation.

1:00 Summary of previous lecture
8:52 Mean squared error (MSE)
16:55 Method of moment estimator
32:02 Maximum likelihood estimator

## Probability and Statistics for Engineers - Exam Questions (Part 1 of 2)

Part 1 of Exam Questions

## Probability and Statistics for Engineers - Exam Questions (Part 2 of 2)

Part 2 of Exam Questions

#probability #statistics

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## ACT Math Prep Study Guide Review

This ACT math prep study guide review youtube video tutorial contains plenty of examples and practice problems with solutions to help you master the concepts that is commonly tested on the act.  It contains tips and strategies to help you some common act math problems in algebra, geometry, and trigonometry.  This video contains the formulas you need to answer very common questions.  This video provides a basic overview of questions you might see on the actual test.  If you need help, you came to the right place.

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## How to Find The Correlation Coefficient in The Linear Relationship

This video explains how to find the correlation coefficient which describes the strength of the linear relationship between two variables x and y.

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## How to Find The Equation Of The Best Fit Line using Linear Regression

This statistics video tutorial explains how to find the equation of the line that best fits the observed data using the least squares method of linear regression.

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## How to Perform A Hypothesis Test Of independence in Statistics

This statistics video tutorial explains how to perform a hypothesis test of independence using the chi-square distribution.

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## Introduction to Chi Square Distribution Test of a Single Variance

This statistics video tutorial provides a basic introduction of the chi square distribution test of a single variance or standard deviation.  It explains how to use it in order to determine whether or not you reject the null hypothesis.

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## Introduction to The Chi Square Test In Statistics

This statistics video tutorial provides a basic introduction into the chi square test.  It explains how to use the chi square distribution to perform a goodness of fit test to determine whether or not to accept or reject the null hypothesis.

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## Introduction Into Matched Or Paired Samples In Statistics

This Statistics video tutorial provides a basic introduction into matched or paired samples.  It explains how to use the T-test and the student's t-distribution to determine whether or not if you should reject the null hypothesis in favor of the alternative hypothesis.  It also explains how to construct a confidence interval and calculate the margin of error at a specified significance level.

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## How to Test Hypothesis with Two Proportions in Statistics With Example

This statistics video tutorial covers hypothesis testing with two proportions.  It provides an example problem that shows you how to determine if the difference between two proportions is significant using the z-test and the normal distribution curve.

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## How to Calculate Cohen's D To Determine If The Size in Statistics

This statistics video tutorial explains how to calculate Cohen's d to determine if the size of the effect is small, medium, or large based on the differences between two sample means.  This video also provides two ways to calculate the pooled standard deviation.

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## How to Perform Hypothesis Testing with Two Sample Means in Statistics

This statistics video explains how to perform hypothesis testing with two sample means using the t-test with the student's t-distribution and the z-test with the normal distribution table.

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## How to Solve Hypothesis Testing Problems with Proportions

This statistics video tutorial explains how to solve hypothesis testing problems with proportions.  It explains how to calculate the sample proportion and the z-test statistic and how to compare that with the critical values in order to determine whether or not if you should accept or reject the null hypothesis at a specified confidence level or significance level.

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## How to Use The P-value To Solve Problems with Hypothesis Testing

This statistics video explains how to use the p-value to solve problems associated with hypothesis testing.  When the p-value is less than alpha, you should reject the null hypothesis and vice versa.  This video discusses when you should use a one tailed test compared to a two tailed test.  It contains two example problems that illustrates how to use the p-value method to determine if you should reject or not reject the null hypothesis.

#statistics  #probability #maths #mathematics

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## Statistics Tutorial: Hypothesis Testing Problems

This statistics video tutorial provides practice problems on hypothesis testing.  It explains how to tell if you should accept or reject the null hypothesis.  It gives two examples - one involving the z-test and the other involving the t-test or t-statistic.  One of those examples include a one tailed test and the other contains an example of the two tailed test.

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## How to Calculate Test Statistic for Means and Population Proportions

This statistics video tutorial provides the formulas for calculating the test statistic for the population mean and population proportion.  This includes the z-statistic and t-statistic.