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An introduction to probability distributions, focusing on random variables, discrete and continuous distributions, and the binomial distribution. It covers the concepts of random phenomena, sample space, probability of an outcome, and the mean and standard deviation of a probability distribution. The document also explains how to find probabilities for bell-shaped distributions and binary data.

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Download Understanding Probability Distributions: Random Variables, Discrete & Continuous, Binomial and more Study notes Statistics in PDF only on Docsity! Chapter 6 Probability Distributions 6.0 Probability Basics A. Random Phenomenon/Activity An activity with possible outcomes; cannot predict with certainty which outcome will result. B. Sample Space The set of possible outcomes of a random activity C. Probability of an Outcome in a Sample Space. A measure of the likelihood (0 - 1 or 0% - 100%) that the outcome will result when the activity is performed. Larger values mean greater likelihood; smaller values mean smaller likelihood. D. Examples • – Random Activity: Toss a fair coin - observe upface – Sample Space = {H,T } – Assuming fair coin, P[H] = 0.5, P[T] = 0.5 – If outcomes in sample space are equally likely, the probabilities are identical, equal to 1 di- vided by number of outcomes 1 2 • – Random Activity: Toss a fair die - observe number of dots on upface. – Sample Space = {1,2,3,4,5,6} – Assuming fair die, P[1] = P[2] = P[3] = P[4] = P[5] = P[6] = 1/6 • – Random Activity: Drop a thumbtack - ob- serve whether point land up or down – Sample Space = {up, down} – Probabilities: Cannot get by equally likely ar- gument, must drop thumback large number of times. Then P[up] = fraction of times point lands up, P[down] = fraction of times lands down. Here probabilities are obtained by observa- tion, repeating random activity large number of times. • – Random Activity: Select JMU student at ran- dom and determine if student is vegetarian (V) or not (NV). – Sample Space = {V,NV} – Outcomes V,NV not equally likely. P[V] = fraction of vegetarians in population, unknown. P[NV] = fraction of non-vegetarians in popu- lation. – When selecting a simple random sample of size n = 1 from a population, then proba- bilities of possible outcomes are fractions in population 5 • The standard deviation σ is the standard devi- ation of a large number of observations of the random variable X – thus it is referred to as a population standard deviation. The standard deviation s = √ Σ(x−x)2 n−1 from Chapter 2 would be the standard deviation of a small number or sam- ple of observations on the random variable X – thus it is now called a sample standard devia- tion. D. Probability Distributions for Continuous Random Vari- ables • Page 253. A continuous random variable has possible values that form an interval. Its prob- ability distribution is specified by a curve that determines the probabilities that the random vari- able falls in any particular interval of values: – Each interval has probability between 0 and 1. This is the area under the curve, above that interval – The interval containing all possible values has probability equal to 1, so the total area under the curve equals to 1. • Example 6 6.2 How Can We Find Probabilities for Bell Shaped Distributions A. The Normal Probability Distribution • Symmetric, bell-shaped, characterized by its mean µ and standard deviation σ. • Probability within any particular number of stan- dard deviations of µ is the same for all normal distributions (0.68 within 1 standard deviation, 0.95 within 2 standard deviations, and 0.997 within 3 standard deviations • Standard Normal Probability Distribution - nor- mal probability of z -scores, where z = (x−µ)/σ; B. Finding probabilities about z-scores - Table A, page A1, A2 in the appendix C. Finding probabilities about more general normal ran- dom variables, X • Examples 7 6.3 How Can We Find Probabilities When Each Observation Has Two Possile Outcomes? A. The Binomial Distribution: Probabilities for Counts with Binary Data • Binary observation Observation that takes on one of two possible outcomes • Random phenomenon: observing n cases or trials of binary observations Summary = number or proportion of n binary cases or trials with outcome of interest. Under certain CONDITIONS the number with outcome of interest has the binomial distribution. • Binomial Conditions – Random phenomenon has n trials, each with two possible outcomes. Outcome of interest is called a “success” and the other outcome is called a “failure”. – Each trial has the same probability of success, denoted by p. – The probability of failure is denoted by 1− p. – The n trials are independent. That is, the result for one trial does not depend on the results of other trials. • Binomial Random Variable X = number of suc- cesses in the n trials for binomial conditions. • Example - Binomial conditions, Binomial Ran- dom Variable 10 • Sampling distribution of sample proportion is ap- proximately normally distributed if np ≥ 15 and n(1 − p) ≥ 15 Example: D. Standard Error • The standard deviation of a sampling distribu- tion is called a standard error 6.5 How Close are Sample Means to Population Means? A. The Sampling Distribution of the Sample Mean • Sample Mean X as a (random) variable • Sampling Distribution of the Sample Mean X. Example (when population sampled is small) 11 B. Mean and Standard Deviation of the Sampling Dis- tribution of X • Suppose random sampling from a population with mean µ and and standard deviation σ. • Mean of the sampling distribution of X is equal to population mean µ. • Standard Deviation of the sampling distribution of X is σ/ √ n (assumes sample size is small rela- tive to pop. size) C. Getting Probabilities about X • Normal Populations If the population is normally distributed, then the sampling distribution of X is normal, regard- less of the size of n • Non-normal populations (Central Limit Theorem) Even if the population is not normal, then the sampling distribution of X is approximately nor- mal for “large” n (Rule of Thumb: n ≥ 30) Example: 6.58, page 297 6.6 How Can We Make Inferences about a Pop- ulation? A. Three Types of Distributions • Population Distribution - see page 298 • Data Distribution - see page 298 • Sampling Distribution - see page 298 12 B. Examples of Three Types of Distributions • Categorical Population (S/F), see Example 22 • Quantitative Population C. Standard Error used to determine how close sample prediction is to population parameter