# Final Exam Note Sheet.docx - Descriptive...

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Descriptive Statistics -Frequency Histogram Bar height = # of observations in each bin (absolute frequency) ● Su m of bar heights = sample size - Relative Frequency Histogram Bar Height (proportion, percentage, relative frequency) =Frequency/Sample Size Example: Bin 1 height = (3/20) = .15 Sum of Bar height = (20/20) = 1 Density Histogram Bar Height (density) = relative freq/bin width Bar area = relative frequency Example: Bin 1 height = . 15/100 = . 0015 Sum of bar area = 1 Larger sd on graph: median~mean Numerical Summary Measures -Location (central tendency, center, average) Median : Rank based central value Quartiles Even: the median, find the medians of the two sets Odd: median as the center value and use that value in both sides Mean : Arithmetic average Notations:Population mean (μ), Sample mean (ȳ), Other cases (E(Y)) - Properties of Frequentist Probability P(A) - probability of event A = (#of outcomes in A)/(total # of different outcomes) 0<=P(A)<=1 P(A) = 1: event A always happens P(A) = 0: event A never happens Addition Rule P(A or B) = P(A) + P(B) if no overlapping (disjoint) Complementary events P(not A) = 1- P(A) -Independent Events : The occurrence of the first event does not change the probability of the occurrence of the second event Product rule P(A and B) = P(A) * P(B) -Bernoulli Distribution Call a RV a Bernoulli RV if it can only realize to the values 0 or 1, and the probability that it realizes to 1 is called π . A Bernoulli RV X can be denoted as X~Bern( π ) If X~Bern( π ) then E(X)= π and VAR(X)= π (1- π ) - Binomial Random Process (Discrete RV) 1. Trials: The random process consists of n identical sub- processes 2. Success vs Failure: each trial results in one of two possible outcomes 3. π : The success rate for every trial is the same 4 . The trial are independent ex: flipping a fair coin four times (n = 4, π = .5) Binomial RV and its pmf B ~ Bin(n, π) = total num of successes achieved in n trials of a binomial random process with probability π of success on any trial b ranges from 0 to n Expectation: E(B) = n π Variance: VAR(B) = n π (1- π ) b (pmf) p(b) B=0,1,...,n n! b! ( n b ) ! π b ( 1 ESTIMATION -Distributions of Functions of RV’s A collection of RVs X1, X2, ..., Xn are said to be independent and identically distributed ( iid) if They are all independent from one another. The realization of any one of them doesn’t change the prob. distribution of any other one They all have exactly the same probability distribution - Standard/Non-Standard Normal Distribution If X is a normal RV, the pdf of X is f ( x )= 1 A normal RV X is denoted X~N(µ,σ 2 ) E(X)=µ, VAR(X)=σ 2 , SD(X)=σ If X~N(µ,σ 2 ), then Z=(x-µ)/σ ~ N(0,1) If Z~N(0,1), then X=Zσ+µ ~ N(µ,σ 2 ) The z such that P(Z>=z)=α, is called z α This is the z critical value -Estimation Rules for Expectation & Varience E(c) = c E(c*X) = c*E(X) E(X+c) = E(X) + c E(X+Y) = E(X) + E(Y) VAR(c) = 0 VAR(c*X) = c 2 VAR(X) VAR(X+c) = VAR(X) If X & Y are independent: VAR(X+Y) = VAR(X) + VAR(Y) Sample Mean: ^ μ = X = Σ X i n
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