Quiz 2 (Unit II) Study Guide

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Quiz 2 is on the material of Unit II (though of course your MATLAB^® knowledge should be cumulative). In preparation for Quiz 2 you should find the study guide/summary below useful.

MATLAB

Double-index arrays (assignment, indexing, dotted arithmetic operators). User-defined functions (syntax, inputs, outputs).The MATLAB built-ins "zeros", "eye", and "ones" and in particular "rand" and "randn".

Text pages to review: Section 5.3 and Sections 6.1–6.4. Also review MATLAB Exercises for Recitation 3 and Recitation 4.

Probability

Frequentist view of probabilities and probability distributions. Mutually exhaustive events, mutually exclusive events, independent events, and corresponding probability "rules." Random variables (discrete and continuous). Mean, variance, and standard deviation.

The Bernoulli discrete random variable (r.v.): 0 and 1 outcomes; probability mass function; meaning of the parameter $\theta$; interpretation as coin flips; mean and variance and standard deviation as a function of $\theta$.

The binomial random variable: definition as a sum of $n$ independent Bernoulli coin flips; outcome tables and probabilities as in examples 9.3.1 and 9.3.2 of the text (in the text X is the Bernoulli r.v., whereas in lecture we used B for the Bernoulli r.v.); shape of the probability mass function (e.g., Figure 9.9 of text).

The univariate and bivariate uniform (continuous) random variable: probability density; interpretation of probability as relative area; generation over any interval or (by independence) rectangle using the "rand" function and affine transformation.

The univariate normal density (continuous) random variable: (see examples 9.4.2 and 9.4.5 of the text); shape and properties; mean $\mu$ and standard deviation $\sigma$; probability that a normal r.v. will take on values within $m \sigma$ of the mean (for m = 1,2,3); generation of normal r.v. realizations for any $\mu$ and $\sigma$ using the "randn" function and affine transformation.

Estimation and Monte Carlo

Bernoulli estimation: sample mean estimate $\hat{\theta}_n$ for the parameter $\theta$; confidence intervals [lower limit, upper limit] for $\theta$; confidence level $\gamma$ and $z_\gamma$ values for 0.80 and 0.95 confidence levels; confidence interval Half Length; Relative Error; dependence on sample size "n". Area estimation: random darts from uniform distribution over rectangle R; "geometric" interpretation; relation between A_D, A_R, and a Bernoulli parameter $\theta$; confidence intervals, Half Lengths, and RelErr for A_D. Good review material: the slides "Bernoulli/Area Estimation Summary" (PDF).

Estimation for the normal density: sample mean estimate $\overline{w}_n$ for $\mu$; sample standard deviation ($s_n$) estimate for standard deviation $\sigma$. (Note on Quiz 2 you will NOT be responsible for confidence intervals for the mean $\mu$ of a normal random variable.) Good review material: "Appendix to Unit II — Estimation: the Normal Density."