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# INFO 1020 Exam 1 Study Guide INFO 1020

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This 8 page Study Guide was uploaded by Alexandra Tilton on Friday January 15, 2016. The Study Guide belongs to INFO 1020 at University of Denver taught by Ray Boersema in Winter 2016. Since its upload, it has received 41 views. For similar materials see Analytics II: Statistics and Analysis in Information System at University of Denver.

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Date Created: 01/15/16

INFO 1020:
Analytics II Exam 1 Study Guide Terms and Deﬁnitions • Experiment: Any process which has a well-deﬁned set of outcomes •Probability: Anumber between and including 0 and 1. It indicates the frequency with which something happens. Can be expressed as a percent, fraction, or decimal • Subjective: Personal opinion of the likelihood of something happening • Empirical: Doing an experiment repeatedly and counting outcomes to determine probability as observed results (rolling dice or tossing coins) • Classical: Mathematical and logical analysis Event: Any subset of the sample space • • Complement: Denoted as A’ and is the event that contains outcomes not in A. P(A) + P(A’) = 1 • Union: Union of two events, B ∪ C, is the set of outcomes belonging to B OR C Alexandra Tilton Page▯ of ▯8 • Intersection: Intersection of two events, B ∩ C, is the set of outcomes belonging to BOTH B AND C • Addition Law: P(B) + P(C) - P(B and C) • Random Variable: “x” whose value is a number which is dependent on the outcome of some experiment or observation •Example: Roll 2 dice • Random Variable: Let x be the distance between the dice • Discrete Random Variable: x-values which are listable. They do not have to be whole numbers. •Example: Number of cars in a parking lot • Continuous Random Variable: x-values which are un-listable. They do not have to be whole numbers. •Examples: temperature, age, time, distance • Probability Function: Aformula or a graph which provides all the x-values and associated P(x) - values •formula: P(x) = 1/2x Ɐ x ϵ {1,2,3,6} • 2 Requirements: • 1: Every x-value •table: x P(x) must have a P(x) 1 1/2 value 2 1/4 • 2: ΣP(x) = 1 3 1/6 6 1/12 •graph: x on x-axis and P(x) on y-axis Alexandra Tilton Page▯ of ▯8 • Uniform Discrete Probability Function: Every P(x) is the same Example: • x P(x) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 • Expected Value of X: The average x, ▯, = Σx⋅P(x) • Variance (VAR) of X: (x-E(x))² ⋅ P(x) • Standard Deviation of X: = square root of variance. This is how far from the mean you have to go to ﬁnd the next data point Types of Probability Distributions • General: P(x)s are different •Formulas: ΣP(x) = 1 • • E(x) = ▯ • VAR(x) = σ² = Σ(x-E(x))² ⋅ P(x) • STDEV(x) = σ = Square root (Σ(x-E(x))² ⋅ P(x)) • Uniform: All P(x)s are the same • Binomial Probability Distributions •Binomial Trial: Any trial that has just two outcomes • The trial is repeated n times • Trials are independent • Two trial outcomes: success and failure Alexandra Tilton Page▯ of ▯8 • Probability of success, p, is constant for each trial • The random variable, x, is the total possible number of successes • x ϵ [0,n] •Example: • Let the trial be toss a die • Let n = 4 • Let success be “1” • Let failure be “not 1” • p = 1/6 • Probability Distribution (List the Xs and the P(X)) • Probabilities formula: P(x) = nCx (p^x) (1-p)^n-x •On calculator: Math, Probability, Combination •Example: P(3) =4C3(1/6)³ (5/6)¹ = 0.0153 • Probability via Excel =BINOM.DIST(x,n,p,0) • • Shortcuts to means and variances • E(x) = ▯ = np = 4(1/6) = .667 • VAR(x) = σ² = np(1-p) = 4(1/6)(5/6) = .5556 • STDEV(x) = σ = sqrt (np(1-p) = 4(1/6)(5/6)) = .74 • Poisson Probability Distributions • Properties •Some events exist •Atime period exists •The random variable, x, is the number of possible events that occur during a given time period •E(x) = ▯ is given •x ϵ [o, ∞] • Example: •Observation: Event: Student coughs •Time period: one minute Alexandra Tilton Page▯ of ▯8 •x will be the number of coughs from 11:25 to 11:26 •E(x) = ▯ = 5 (this is given to us) •Calculate probability of 3 coughs • Probability formula: (▯ )(e )/x! • Probability via Excel •=POISSON.DIST(x,mean,0) •IF PROBABILITY (P(X)) IS LESS THAN .001, STOP TABLE •If you get a problem that asks what the probability of getting something greater than the probability you stopped at, then add up all until that number and subtract from 1. • Shortcuts to means and variances • E(x) = ▯ = given • VAR(x) = σ² = same as E(x) • STDEV(x) = σ = sqrt (E(x)) • Hypergeometric Probability Distributions • Properties Apopulation of size N (N objects) • •There is a subset of the population and its size is r. These could be called successes •Trial = pick n objects from the pot •The random variable, x, is the number of successes •x ϵ [0,n] •The random variable, x, is the total possible number of successes •x ϵ [0,n] or x ϵ [0,(whichever is smaller) • Example: •N = 52 (cards) •r = 12 (faces) •n = 5 •x = # of face cards drawn Alexandra Tilton Pag▯ of ▯8 • Probabilities formula: P(x) = (rCx) (N-rCn-x) (NCn) • Probability via Excel •=HYPGEOM.DIST(x,n,r,N,0) • Shortcuts to means and variances •***p = r/n (12/52) •E(x) = ▯ = np •VAR(x) = σ² = np(1-p) •STDEV(x) = σ = sqrt (np(1-p) = 4(1/6)(5/6)) = .74 • Properties of all Continuous Probability Distributions • a: x ϵ : Real numbers •not listable • b: There may or may not be a smallest x, there may be a largest x • c: There is a probability function • d: The area under the graph is 1 • e: Areas are probabilities • f: P(x a particular number) = 0 • g: All probabilities will be for intervals of x-values •inclusive doesn’t make a difference • Properties of Uniform Continuous Probability Distributions • a: There IS a smallest x-value “a” •There IS a largest x-value “b” • b: y=f(x)=k (constant) •Straight line from “a” to “b” •Length of box (on graph) = “b”-“a” •Hight of box = 1/(“b”-“a”) Alexandra Tilton Page▯ of ▯8 • c: P(c<x<d) = box within box = length*width •Area is (d-c)(1/b-a) • Example: Let x be time •Let a = 0 •Let b = 58 •P(X>40) = (18)(1/58) = 18/58 • d: The E(x) = ▯ = a+b/2 • e: The VAR(x) = σ² = (b-a)²/12 • f: σ = Square root of the VAR(x) • Properties of Standard Normal Continuous Probability Distributions • a: The random variable is z (not x) • b: No smallest z value (negative inﬁnity) •No highest z value (positive inﬁnity) • c: E(z) = 0 • d: STDEV(z) = 1 • e: y=f(x) is bell-shaped, goes from -3 to +3 • f: y=f(x)=(1/square root of (2pi))e^(-x²/2) = probability curve • g: Remember the Empirical Rule (.68), (.9544), (.9973) • EXCEL: =NORM.S.DIST(1,1) •P(Z<1.72): =NORM.S.DIST(1.72,1) •P(Z>1.72): =1-NORM.S.DIST(1.72,1) •P(-1.6<z<1.8): =NORM.S.DIST(1.8,1) - =NORM.S.DIST(-1.6,1) Alexandra Tilton Page▯ of ▯8 • h: To ﬁnd a cut point (z-value) =NORM.S.INV(area to the left) (subtract from one if you’re given it the other way around • Properties of Normal Continuous Probability Distributions • a: The random variable is x • b: No smallest x value •No largest x value • c: The E(x) = ▯ = given • d: σ = Square root of the VAR(x) = given • e: y=f(x) is bell-shaped, goes from ▯-3σ to ▯+3σ • EXCEL: =NORM.DIST(x value, mean, stdev,1) •Subtract from one if you need greater than (to the right) •Subtraction from two points if you need between CUT POINT =Norm.INV(area to the left, mean, stdev) • Alexandra Tilton Pag▯ of ▯8

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