Motivation Suppose you have a population with known \(\mu\) and \(\sigma\). You then take a sample (perhaps not randomly) and discover...
For a Poission distribution with large \(n\), use \(Z=\frac{\bar{X}-\lambda}{\sqrt{\lambda/n}}\) (i.e. \(\lambda\) instead of \(S\)). It...
Things you should think about: What are the implications of your choice of \(\alpha\)? How much did intuition play a role in deciding...
The p-value is the smallest significance level (i.e. \(\alpha\)) at which \(H_{0}\) would be rejected. So if \(p\le\alpha'\), you reject...
Let \(X\) be the number of successes. If \(n<
Steps to Carry Out The Experiment Identify the parameter of interest. Determine the null value and state the null hypothesis. State the...
The null hypothesis (\(H_{0}\)) vs the alternative hypothesis (\(H_{a}\)): The null hypothesis is assumed to be true (default). The...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\) If \(\Sample\) be...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\) Let \(\Sample\) be...
The one-sample t-distribution confidence interval is robust for small or even moderate departures from normality, unless \(n\) is very...
What if you have \(n\) observations in a normal distribution and want to predict \(X_{n+1}\)? The prediction interval is \(\bar{x}\pm...
Say you take a sample where \(n\) is not large. Then the CLT doesn’t apply. We must then know/assume a distribution. Suppose we know it...
Let \(\Sample\) be independent and identically distributed from \(N(\mu,\sigma^{2})\). Define the following random variable:...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\) For any...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\) Assume you have a...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\) A good estimate...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) \(\newcommand{\Sample}{X_{1},\dots,X_{n}}\) The Method...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) Notation: \(\hat{\mu}=\bar{X}\) means the point estimator of...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) Let \(X_{1}\dots X_{n}\) have means \(\mu_{i}\) and variances...
Let \(X_{1}\dots X_{n}\) be a random sample from a distribution with mean \(\mu\) and standard deviation \(\sigma\). Then:...
When we take a sample and calculate its mean and standard deviation, this is treated as a random variable for the population...
\(\newcommand{\Cov}{\mathrm{Cov}}\) \(\newcommand{\Corr}{\mathrm{Corr}}\) Expected Value The expected value of a function \(h(X,Y)\) is...
Probability Given two random variables \(X,Y\), the joint pdf is given by \(p(x,y)=P(X=x,Y=y)\). Let \(A\) be an event. Then the joint...
Sample Percentiles Calculating sample percentiles is challenging. What is the 23rd percentile of 10 points? One rule: Order the \(n\)...
For Weibull, let \(Y=\ln(X)\). This has both scale and location parameters. Location is \(\theta_{1}=\ln(\beta)\) and scale is...
The beta distribution The parameters are \(\alpha,\beta>0\), and \(A,B\) with \(B\ge A\). \begin{equation*}...
If \(Y=\ln X\) is a normal distribution, then \(X\) is log-normal. \begin{equation*} f(x;\mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma}...
Probability Density Function Let \(\alpha,\beta>0\) \begin{equation*}...
If the time between successive events is independent each with an exponential distribution with \(\lambda\), then the total time \(X\)...
Chi-squared distribution Probability Density Function The parameter is \(\nu\) and it is a positive integer. It is the gamma...
Exponential distribution Probability Density Function Let \(\lambda>0\) \begin{equation*} f(x;\lambda)=\lambda e^{-\lambda x}...
The problem with the normal distribution is that it is symmetric. The Gamma distribution is useful for skewed distributions....
Probability Distribution Function The parameters are \(-\infty<\mu<\infty\) and \(\sigma>0\). \begin{equation*}...
The Pareto distribution is good for approximating income distributions or population sizes. The pdf is given by: \begin{equation*}...
Continuous distributions are given by a probability density function (pdf): \begin{equation*} P\left(a\le X\le...
Suppose you have a library with \(M\) items and you want to sort them by popularity. The parameters are \(M\) and \(\alpha\). The domain...
\(X\) is of a Poisson distribution if its pmf is \(p(x;\lambda)=\frac{e^{-\lambda}\lambda^{x}}{x!}\) where \(x\) is 0, 1, 2, etc....
The experiment requires: The trials be independent The outcome is binary (success or failure). The probability of success or failure is...
The hypergeometric experiment requires: The population is finite, with \(N\) individuals. The outcome of each trial is binary (success...
Bernoulli Distribution A Bernoulli random variable is one whose only possible values are 0 and 1. \(P(X=1)=p\) \(E[X]=p,V[X]=p(1-p)\)...
Random Variables A discrete random variable is one whose set of possible values is countable. Probability Distributions for Discrete...
Read hypothesis testing in Chapter 2 (ML rules). Reliability in Chapter 2
Sample Spaces and Events The sample space \(\mathcal{S}\) is a set. An event is a subset of the sample space. A simple event is an event...
Measures of Location When reporting a sample mean, use one extra significant digit. Sample mean: \(\bar{x}\) Population mean: \(\mu\)...
Pictorial and Tabular Methods in Descriptive Statistics Stem and Leaf Display Stem and leaf display Advantages: Useful for displaying...
Populations, Samples and Processes A census means you poll every member of the population. Univariate vs bivariate or multivariate:...
Linearity of expectation: \(E(aX+bY+c)=aE(x)+bE(Y)+c\). This holds even when \(X\) and \(Y\) are dependent. Exploit this! Independent...