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Descriptive Statistics-Frequency Histogram●Bar height = # of observations in each bin (absolute frequency)●● Sumof barheights =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) = 1Density Histogram●Bar Height (density) = relative freq/bin width●Bar area =relativefrequency●Example:Bin1 height =.15/100 = .0015●Sum ofbararea = 1Larger sd on graph: median~meanNumerical Summary Measures-Location (central tendency, center, average)●Median: Rank based central value ●Quartiles○Even: the median, find the mediansof 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 ProbabilityP(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)=πand VAR(X)=π(1-π)-Binomial Random Process (Discrete RV)1. Trials:The random process consists of n identical sub-processes2. Success vs Failure: each trial results in one of two possible outcomes3. π: The success rate for every trial is the same4. The trial are independentex: 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) = nπ●Variance: VAR(B) = nπ(1-π)b (pmf)p(b)B=0,1,...,nn!b!(n−b)!πb(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,σ2)○E(X)=µ, VAR(X)=σ2, SD(X)=σ●If X~N(µ,σ2), then Z=(x-µ)/σ ~ N(0,1)●If Z~N(0,1), then X=Zσ+µ ~ N(µ,σ2)●The z such that P(Z>=z)=α, is called zα○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) = c2VAR(X)○VAR(X+c) = VAR(X)○If X & Y are independent:○VAR(X+Y) = VAR(X) + VAR(Y)●Sample Mean: ^μ=X=Σ Xin●