Introduction to random variables

Jan 20, 20096 likes6,723 views

This document provides an introduction to random variables. It defines random variables as functions that assign real numbers to outcomes of an experiment. Random variables can be either discrete or continuous depending on whether their possible values are countable or uncountable. The document also defines probability mass functions (pmf) which describe the probabilities of discrete random variables taking on particular values. Expectation is introduced as a way to summarize random variables using a single number by taking a weighted average of all possible outcomes.

This document provides an overview of key concepts related to random variables and probability distributions. It discusses: - Two types of random variables - discrete and continuous. Discrete variables can take countable values, continuous can be any value in an interval. - Probability distributions for discrete random variables, which specify the probability of each possible outcome. Examples of common discrete distributions like binomial and Poisson are provided. - Key properties and calculations for discrete distributions like expected value, variance, and the formulas for binomial and Poisson probabilities. - Other discrete distributions like hypergeometric are introduced for situations where outcomes are not independent. Examples are provided to demonstrate calculating probabilities for each type of distribution.

Discrete Random Variables And Probability Distributionsmathscontent

1. The document defines discrete random variables as random variables that can take on a finite or countable number of values. It provides an example of a discrete random variable being the number of heads from 4 coin tosses. 2. It introduces the probability mass function (PMF) as a function that gives the probability of a discrete random variable taking on a particular value. The PMF must be greater than or equal to 0 and sum to 1. 3. The cumulative distribution function (CDF) of a discrete random variable is defined as the sum of the PMF values up to that point. It ranges from 0 to 1 and increases monotonically.

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This document introduces key concepts related to random variables and probability distributions: - A random variable is a function that assigns a numerical value to each possible outcome of an experiment. Random variables can be discrete or continuous. - A probability distribution specifies the possible values of a random variable and their probabilities. For discrete random variables, this is called a probability mass function. - Key properties of a probability distribution are that each probability is between 0 and 1, and the sum of all probabilities equals 1. - The mean, variance, and standard deviation can be calculated from a probability distribution. The mean is the expected value, while variance and standard deviation measure dispersion around the mean.

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This document defines key concepts related to random variables including: - A random variable is a numerical measure of outcomes from a random phenomenon. - Probability distributions describe the probabilities associated with random variables. - Expected value refers to the mean or weighted average of a probability distribution. - As the number of trials increases, the actual mean approaches the true mean due to the Law of Large Numbers. - Binomial and geometric distributions model situations with success/failure outcomes and independence between trials.

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This document discusses basic concepts of probability, including: - The addition rule and multiplication rule for calculating probabilities of compound events. - Events can be disjoint (mutually exclusive) or not disjoint. - The probability of an event occurring or its complement must equal 1. - How to calculate the probability of at least one occurrence of an event using the complement. - When applying the multiplication rule, you must consider whether events are independent or dependent.

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Random variables can be either discrete or continuous. A discrete random variable takes on countable values, while a continuous random variable can take on any value within a range. The probability distributions for discrete and continuous random variables are different. A discrete probability distribution lists each possible value and its probability, while a continuous distribution is described using a probability density function. Random variables are used widely in statistics and probability to model outcomes of experiments and random phenomena.

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Basic concept of probabilityIkhlas Rahman

This presentation provides an introduction to basic probability concepts. It defines probability as the study of randomness and uncertainty, and describes how probability was originally associated with games of chance. Key concepts discussed include random experiments, sample spaces, events, unions and intersections of events, and Venn diagrams. The presentation establishes the axioms of probability, including that a probability must be between 0 and 1, the probability of the sample space is 1, and probabilities of mutually exclusive events sum to the total probability. Formulas for computing probabilities of unions, intersections, and complements of events are also presented.

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The document discusses exploratory data graphics and principles for critiquing graphics. It introduces the "four C's" framework for critique: content, construction, context, and consumption. It also discusses the grammar of graphics as an abstraction for reasoning about and communicating graphics, focusing on the key components of layers, data mappings, geoms, and statistics. Multiple examples are provided and attendees are prompted to practice converting between textual descriptions and ggplot code.

15 time-spaceHadley Wickham

The document discusses visualizing time series data on baby names by state in the US from 1960-2008. It demonstrates plotting trends in a name's popularity over time both nationally and by individual state, and explores techniques like smoothing, centering, and scaling the data to better analyze patterns. Maps showing the geographic distribution of a name's popularity in different states in also presented.

14 case-studyHadley Wickham

This document discusses analyzing name usage data from a dataset containing boy and girl name usage over time. It uses the plyr package to summarize a subset of names that are used for both boys and girls, looking at the number of years each name has been used and its average usage. Graphs are created to examine patterns in the log ratio of boy to girl usage for selected names over 50 years that had average usage over 0.0005. The analysis aims to understand how usage of these unisex names has changed over time and to potentially estimate error rates in the data.

13 case-studyHadley Wickham

This document discusses a case study analyzing gender trends in baby names: 1. It focuses on analyzing a smaller subset of the most popular names over time. 2. The document computes summary statistics like the proportion of babies given each name that were boys or girls and the number of years each name was in the top 1000 for boys and girls. 3. Names are classified as either having dual or separate gender usage over time based on the ratio of boy to girl name assignments each year.

12 adv-manipHadley Wickham

11 adv-manipHadley Wickham

1. Split the data frame into pieces by name using ddply() or split(). 2. Apply a function like summarise() to each piece to calculate the sum of people with that name. 3. Combine the results into a new data frame with the total for each name. This allows grouping operations like calculating a summary statistic for each unique value in a variable. The data is split, a function is applied to each group, and then combined into a single data frame with the results.

11 adv-manipHadley Wickham

This document discusses advanced data manipulation techniques in R using packages like plyr and ggplot2. It covers slicing and dicing data frames using functions like subset, summarise, transform and arrange. It also discusses merging data from multiple sources and performing group-wise operations on data. Brainstorming ideas include exploring trends in baby names data over time by variables like first letter, length, rankings, and merging with additional data sources.

10 simulationHadley Wickham

This document discusses simulation and random number generation. It covers using floating figures with captions in homework, including comments in code, calculating probabilities and expected values mathematically for simulations, generating random numbers from different distributions using functions like runif and rnorm, pseudorandom number generation in computers using linear congruential generators, and sources of true randomness like atmospheric noise.

27 developmentHadley Wickham

24 modellingHadley Wickham

23 data-structuresHadley Wickham

Graphical inferenceHadley Wickham

R packagesHadley Wickham

22 spamHadley Wickham

21 spamHadley Wickham

20 date-timesHadley Wickham

19 tablesHadley Wickham

18 cleaningHadley Wickham

17 polishingHadley Wickham

16 critiqueHadley Wickham

15 time-spaceHadley Wickham

14 case-studyHadley Wickham

13 case-studyHadley Wickham

12 adv-manipHadley Wickham

11 adv-manipHadley Wickham

10 simulationHadley Wickham

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