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Probability and Statistics Study.docx

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Department
Economics
Course
ECO3150
Professor
Qi Zhang
Semester
Winter

Description
Chapter 1 Describing Data: Graphical Ben Zakaib P(A |B )P(B ) F(x )=P(X ≤ x ) F(x0)=P(X ≤ x )0 ∑ P(x) =1 P(1 |1 ) =1 1 1 P(B 1A 1 > P(B 1 0 0 x P(1 ) P(B |A ) P(B ) and 2 1 2 P(1 1A )1(A ) P(A1|1 ) = P(B ) 1 P(A 1B 1 P(A) P(A) odds = = P(A 1B 2 1-P(A) P(A) number of outcomessatisfying Aand B P(A ∩B ) P(A ∩ B ) P(A∩B)= i j i j totalnumber of elementary outcomes P(B | A)=P(B) P(A | B)=P(A) P(A ∩ B)=P(A)P(B) P(A ∩B)=P(B|A)P(A) P(A ∩B)=P(A|B)P(B) P(A∪B)= P(A)+P(B)−P(A∩B) P(A)=1−P(A) P(A∪B)=P(A)+P(B) P(A)= P(O ) 0 ≤ P(A)≤1 n P n ∑A I P(A) = A C = x n x x! n n! cx= x!(n− x)! n! x(x−1)(x−2)...(2)(1)= x! n! N N P = n A P(A)= A x c k N (n− x)! k!(n−k)! P n Cx= x x! n! Statistics – a tool to help process, summarize, analyze and interpret data cx= x!(n− x)! Population (N) – the complete set of All Items that interest an investigator Characteristics of a population N Represents the population size N can be very large or even infinite Sample (n) – an observed subset of the population Characteristics of a Sample n represents the sample size Sources of Errors Sampling errors – information is availiable on only a subset of all the population Non sampling Errors – the population sampled is not the relevant one; survey respondents may give inaccurate answers; no responses to survey questions Classification of Variables Categorical variables – produce responses that belong to groups or categories Numerical variables Discrete – finite number of values; from counting process (e.g. number of students) Continuous – any value within a given range of numbers; from measurement process (e.g. distance, temperature) Measurement levels Qualitative – no measurable meaning Nominal data – categories; numbers have no meaning Ordinal – ranking Quantitative – measurable meaning Interval – Distance from an arbitrary zero Ratio – distance from a natural zero Bar Charts Bar!C harts! • Bar!charts:!frequency!(component;!cluster)! Data$ Category$ Number$of$ Students$ Very!good! 29! Good! 55! Moderate! 20! Poor! 9! 22! Bar!C harts! Gender$and$HealthStatus(Component)$ Data$ Category$ Males$ Females$ Very!good! 16! 13! Good! 26! 29! Moderate! 12! 8! Poor! 7! 2! 23! Bar!C harts! Data$ Gender$and$HealthStatus(Cluster)$ Category$ Males$ Females$ Very!good! 16! 13! Good! 26! 29! Moderate! 12! 8! Poor! 7! 2! 24! P ie!C harts! Pie Charts: Proportion • Pie!charts:!propor*on! Table:By$propor- on$ Category$ Number$of$ Propor- on$ Students$ Very!good! 29! 26%! Good! 55! 49%! Moderate! 20! 18%! Poor! 9! 8%! Total! 113! 100%! 25! Pareto Diagrams To identify major causes of problem’s Pareto: Small number of factors are responsible for most of the problems Pareto diagrams: displays the frequency of defect causes Left -> right: most frequent to least frequent (con%nued)* Cross Sectional Vs. Time Series Cross-sectional data: the sample which we collect at one point in time—e.g. Canada’s exports trade flows in 2010 by sectors; Time-series data: data measured at successive points in time—e.g. Canada’s exports Figure:Currency$Exchange$ Rates:CAD$to$USD$(Time$SeriesPlot)$ trade flows from 2000-2012 in manufacture sector Frequency Distribution for Numerical data what… a frequency distribution is a list or a table containing class grouping (categories or ranges within which the data fall) and the corresponding frequencies with which data fall within each class or category determining the width of an interval F requency!Distribu*on!E xample! Frequency Distribution Example Data!in!order!array:! 12! 13! 17! 21! 24! 24! 26! 27! 27! 30! 32! 35! 37! 38! 41! 43! 44! 46! 53! 58! Interval$ Frequency$$ Rela- veFrequency$ Percentage$ 10!but!less!than!20!! 3! 0.15! 15%! 20!but!less!than!30!! 6! 0.3! 30%! 30!but!less!than!40!! 5! 0.25! 25%! 40!but!less!than!50!! 4! 0.2! 20%! 50!but!less!than!60!! 2! 0.1! 10%! Total!! 20! 1! 100%! 39! Histogram A graph of the data in a frequency distribution is called a histogram The interval endpoints are shown on the horizontal axis The vertical axis is either Frequecy, relative frequency, or percentage Bars of the appropriate heightss are used to represent the number of observations within each class Histogram! !Interval! Frequency! 10!but!less!than!20! 3! 20!but!less!than!30! 6! 30!but!less!than!40! 5! 40!but!less!than!50! 4! 50!but!less!than!60! 2! No!gaps!between! bars.! 41! Many vs. Few Many!vs.!F ew! • Many$ (Narrow$ class$intervals)! • may!yield!gaps!from!empty! classes!! • Can!give!a!poor!indica*on!of! how!frequency!varies!across! classes! • Few$ (Wide$ class$intervals)! • may!compress!varia*on!too! much!and!yield!a!blocky! distribu*on! • Hide!important!paRerns!of! varia*on.! 43! The Cumulative Frequency Distribution The!C umula*ve!F requency! Distribu*on! Data!in!order!array:! 24$ 35$ 17$ 21$ 24$ 37$ 26$ 46$ 58$ 30$ 32! 13! 12! 38! 41! 43! 44! 27! 53! 27! Interva$$ Frequency$$ Rela- ve$ Percentage$ Cumula- ve$ Cumula- ve$ Frequency$$ Frequency$ Percentage$ 10!but!less!than!20!! 3! 0.15! 15%! 3! 15%! 20!but!less!than!30!! 6! 0.3! 30%! 9! 45%! 30!but!less!than!40!! 5! 0.25! 25%! 14! 70%! 40!but!less!than!50!! 4! 0.2! 20%! 18! 90%! 50!but!less!than!60!! 2! 0.1! 10%! 20! 100%! Total!! 20! 1! 100%! 44! The!Ogive! The Ogive: Graphing CumGraphing!Cequencumula*ve!Frequencies! Interva$$ Cumula- ve$ Percentage$ 10!but!less!than!20!15%! 20!but!less!than!30!! 45%! 30!but!less!than!40!70%! 40!but!less!than!50!90%! 50!but!less!than!60!! 100%! Interval!endpoints! Ch.!1:45! Stem and Leaf Diagram A simple way to see distribution details in a data set Method: separate the sorted data series Into leading digits (the stem) and the Trailing digits (the leaves) 17$ 21$ 24$ 24$ 26$ 30$ 35$ 37$ 38$ 46$ • " " Relationships between variables Scatter Diagrams Scatter diagrams: used for paired observations taken from two numerical variables One variable is measured on the vertical axis and the other variable is measured on the horizontal axis S caRer!Diagram!E xample! Table:!SAT!Math!vs.!GPA! Figure:!SAT!Math!vs.!GPA!(ScaRer!Plot)! SATMath$ GPA$ 450! 3.25! 480! 2.6! 500! 2.88! 520! 2.85! 560! 3.30! 580! 3.10! 590! 3.35! 600! 3.20! 620! 3.50! 650! 3.59! 700! 3.95! 50! Cross Tables The number of observations for every combination of values for two categorical or ordinal variables If there are (r) for the first variable (rows) and (c) categories for the second variable (columns), the table is called an r x c cross table C ross!Table!vs.!C luster!Bar!C hart! 53! Data presentation Errors Goals for effect data presentation: Present data to display essential information Communicate complex ideas clearly and accurately Avoid distortion that might convey the wrong message Unequal histogram interval widths Compressing or distorting the vertical axis Providing no zero point on the vertical axis Failing to provide relative bass in comparing data between groups Chapter 2: Describing Data: Numerical Ben Zakaib Des cribingDataNumerically Describing Data Numerically Describing Data Numerically Central Tendency Variation Arithmetic Mean Range Median Interquartile Range Mode Variance Standard Deviation Coefficient of Variation Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall4 Arithmetic Mean Mean the most common measure of central tendency Mean = sum of values divided by the number of values For a population of N values: For a sample of size n: The mean can be affect by extreme values (outliers) Median Arrange the data in either increasing or decreasing order; In an ordered list, the median is the “middle” number (50% above, 50% below) • Arrangethedataineither increasingor dec nreasing order; Chapter 2: Describing Data: Numerical Ben Zakaib • Ina orderedlist, themedianh ist e“middle” number (50%above, 50%below) The Median is not affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 the Location of Median = 3 Median = 3 the Median • Not affectedbyextremevalues Copyright © 2013 Pearson Education, Inc. Publishing as Ch. 2-9e Hall Mode Mode • Ameasureof central tendency; Chapter 2:• Valuethat occursmost often; Ben Zakaib A measure of central tendencymevalues; Value that occurs most oftenrical or categorical data; Not affected by extreme values Used for either numerical or categorical data • Theremaybeseveral modes; There may be no mode or several modes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14ib0 1 2 3 4 5 6 Mode = 9 No Mode Copyright © 2013 Pearson Education, Inc. PubliCh. 2-11 Prentice Hall • Describeshow dataaredistributed Shape of a Distribution • Measuresof shape Measures of Shape Symmeti– Symmetricor skewed Left-Skewed Symmetric Right-Skewed Mean < Median Mean = Median Median < Mean Copyright © 2013 Pearson Education, Inc. Publishing asCh. 2-20e Hall Symmetry – the shape of distribution is symmetric if the observations are balanced, or approximately evenly distributed, about its middle Skewness – a distribution is skewed, or asymmetric, if the observations are not symmetrically distributed on either side of the middle Geometric Mean Used to measure the rate of change of a variable over time; The nth root of the product of n numbers Chapter 2: Describing Data: Numerical Ben Zakaib geometric mean rate of return measures the status of an investment over time Arithmetic mean vs. Geometric Mean Measures of Variability Mean is not enough to describe data We need measures to describe the variability of data: range, quartiles, interquartile range; variance, standard deviation, and coefficient of variation Measures of variability – Range Range – the difference between the larges and smallest observations Rule – the greater the spread of the data form the center of the distribution, the larger the range will be Measures the total spread of the data, but may be unsatifactor measure of variability because of outliers Simplest measure of variation • S implest measureof variation Chapter 2: Describing Data: Numerical Ben Zakaib • Differencebetween thelargest and the smallest observations: Range = X largest X smallest Example: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Range = 14 - 1 = 13 35 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Measures of Variability – Quartiles We may need to remove a few of the lowest observations or the highest observations Percentiles and quartiles indicate the position of a value relative to the entire set of data Generally used to describe large data sets P percentile = value located in the (P/100)(n+1) ordered position Quartiles split the ranked data in 4 segments with an equal number of values per segment (note that the widths of the segments bay be different) First quartile position: Q1 = the value in the 0.25(n+1)th ordered position Second quartile position: Q2 = the value in the 0.50(n+1)th ordered position(the median position) Third quartile position: Q3 = the value in the 0.75(n+1)th ordered position where n is the number of observed values Quartile Example Chapter 2: Describing Data: Numerical Ben Zakaib Interquartile Range IQR – measures the spread in the middle 50% of the data; the difference between the observation at Q3, the third quartile, and the observation at q1, the first quartile IQR = Q3 – Q1 Five-Number Summary Refers to five descriptive measures Minimum First quartile Median Third quartile Maximum Minimum < Q1 < Median < Q3 < Maximum Population Variance Average of squared deviations of values fom the mean Population variance: Where: μ = population mean N = population size xi = ith value of the variable x Chapter 2: Describing Data: Numerical Ben Zakaib Sample Variance Average (approximately) of squared deviations of values from the mean Sample variance Where: Xbar = arithmetic mean n = sample size Xi = ith value of the variable X Population Standard Deviation Most commonly used measure of variation Shows variation about the mean population standard deviation: Sample Standard Deviation Most commonly used measure of variation Shows variation about the mean Has the same units as the original data Sample standard deviation Calculation example: sample Standard Deviation Chapter 2: Describing Data: Numerical Ben Zakaib Measuringvariation Measuring variation Small standard deviation Large standard deviation Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Coefficient of Variation Measures relative variation: a measure of relative dispersion that express the standard deviation as a percentage of the mean Always in percentage Shows variation relative to mean Van be used to comparte two or more sets of data measure in different units     Chapter 2: Describing Data: Numerical Ben Zakaib Chebychev’s Theorem For any population with mean μ and standard deviation σ , and k > 1 , the percentage of observations that fall within the interval [μ + kσ] Is at least Regardless of how the data are distributed, at least (1 - 1/k2) of the values will fall within k standard deviations of the mean (for k > 1) The Empirical Rule For many large populations (bell-shaped), this rule provides an estimate of the approximate percentage of observation that are contained with one, two, or three standard deviation of the mean μ +/- 1σ contains about 68% of the values in the population or the sample μ +/- 2σ contains about 95% of the values in the population or the sample μ +/- 3σ contains almost all (about 99.7%) of the value in the population sample Weighted Mean and Measure of Grouped Data The weighted mean of a set of data is where wi is the weigh of the ith observation and to be used when data is already grouped into n classes, with wi value in the ith class Chapter 2: Describing Data: Numerical Ben Zakaib E xample-G PA Example – GPA Semester AcademicRecord CREDITHOURS CREDITHOURS COURSE GRADE wi VALUE xi XVALUEwi*xi English A 3 4 12 Math B 3 3 9 BiologyLabC 4 2 8 Spanish D 5 1 5 Total 15 34 If each course were given the same number of credit hours, the stuent’s e ster GPA would equal? • Csost of purchaseships between Variability frequency midpoint squreof fi*squreof Cost fi mi fi*mi mi-mean (mi-mean) (mi-mean) 0<2 2 1 2 -4.6 21.16 42.32 2<4 3 3 9 -2.6 6.76 20.28 4<6 6 5 30 -0.6 0.36 2.16 6<8 5 7 35 1.4 1.96 659.8 8<10 4 9 36 3.4 11.56 46.24 mean 5.6 sums 112 120.8 • Themean is?elationships Between Variables Covariance a measure of the direction of a linear relationship between two variables Correlation Coefficient a measure of both the direction and the strength of a linear relationship between two variables Covarianceestandard deviation is? 00 Chapter 2: Describing Data: Numerical Ben Zakaib Interpreting Covariance Covariance between two variables Cov(x,y) > 0 x and y tend to move in the same direction Cov(x,y) < 0 x and y tend to move in opposite directions Cov(x,y) = 0 x and y are independent Coefficient of Correlation Chapter 3 Probability Ben Zakaib Probability Probability: how they can be determined, and how they can be used Basic concepts in probability Random Experiment – a process leading to an uncertain outcome Basic Outcome – a possible outcome of a random experiment Sample Space (S) – the set of all possible outcomes of a random experiment Rule 1 – the basic outcomes in such a way that no two outcomes can occur simultaneously Rule 2 – the random experiment must lead to the occurrence of one of the basic outcomes Event (E) – any subset of basic outcomes from the sample space Example – random experiments/ basic outcomes/ sample spaces If a coin is tossed, what are the possible basic outcomes at this random experiment? Solution: the sample space is: S=[{1. head},+{2. tail}] Events and Intersection of Events – if A and B are two events in a sample space S, then the intersection, A.∩.B is the set of all outcomes in S that belong to both A and B The intersection A ∩ B occurs of and only if both A and B occur. Joint probability of A and B to denote the probability of A and B B A and B are Mutually Exclusive Events if they have no basic outcomes in common (conued) Chapter 3 Probability Ben Zakaib Union of Events – If and A and B are two events in sample space S, then the union, A,U,B, is the set of all outcomes in S that belong to either A or B Events UB • T he*C omplement* of*an*event* A: • *Let*A*be* an*event* inthe* sample* space,* S.T he set*of*allbaslicoutcomes*haubelonging* to*S but not* to*A*iscalled*the*complement*d toof*A*islectively exhaustive, if E1 U E2 U … U Ek = S, The Complements of an event A denoted* Let A be an event in the sample space, S. The Set of all basic outcomes belonging to S * but not to A is called the complement of A is denoted S A * S = (1,2,3,4,5,6) A = (2,4,6) B = (4,5,6) 0 Mutually exclusive: A and B are not mutually exclusive The outcomes 4 and 6 are common to both Collectively exhaustive A and B are not collectively exhaustive A U B does not contain 1 or 3 Probability and its Postulates Probability – the chance that an uncertain event will occur (always between 0 and 1) Chapter 3 Probability Ben Zakaib Assessing Probability Classical Probability – assumes all outcomes in the sample space are equally likely to occur Classical probability of event a: where is the number of outcomes that satisfy the conditions of Event A N = total amount of outcomes in the Sample Space Formula for determining the number of combinations The Combinations formula determines the number of combination of (n) items taken (k) at a time Permutations and Combinations The number of possible orderings: The total number of possible way of arranging x objects in order is given by Suppose we have a no. of n object with which the x ordered boxes could be filled (n>x). Each object may be used only once The no. of possible orderings is called the no. of permutations of x objects chosen from n, and denoted as Permutations: the number of possible arrangement when x objects are to be selected from a total of n objects and arranged in order [with (n-x) objects left over] Chapter 3 Probability Ben Zakaib Combination: the number of combinations of x objects chosen from n is the number of possible selections that can be made Relative Frequency The relative frequency probability is the no. of events in the population that meet the condition divided by the total no. in the population These probabilitesi indicate how often an event will occur compared to other events The relative frequency probability is the limit of the proportion of times that even A occurs in a large no. of trials n, n is the number of A outcomes A n is the total number of trials or outcomes Example Example 1 The A and B football teams have played each other 50 times. A has won 10 times, B as won 35 time, and the teams have drawn 5 times We weant to estimate the probability that B will win the next match. So far, B have won 35 out of 50 matches. We can write this as a fraction, which is 35/50 This fraction isn’t the probability of B winning, but it is an estimate of that Probability We say that the relative frequency of B winning is .7 Subjective Probability Chapter 3 Probability Ben Zakaib Subjective probability expresses an individual’s degree of belief about the chance that an event will occur. These subjective probabilities are used in certain management decision procedures Probability Postulates Let S denote the sample space of a random experiment, Oi, the basic outcomes, and A an event. For each even A of the sample space, S, we assume that P(A) is defined and we have the following probability postulatesL If A is the event in the sample space, S, Let A be an event in S, and let Oi denote the basic outcomes. Then… Where the notation implies that the summation extends over all the basic outcome in A P(S) = 1 Consequences of the Postulates 1. If sample space, S, consists of n equally likely basic outcomes, E1, E2,…,En, then… P(Oi)= 1/n i = 1,2,…,n 2. The sample space S consists of n equally likely basic outcomes and event A consists of nA of these outcomes, then P(A) = nA/n 3. A and B are mutually exclusive events. The probability of their union is the sum of their individual probabilities proof: apply postulate 2 P(A∪B) = ∑ P(O i A∪B where Chapter 3 Probability Ben Zakaib ∑ P(Oi) =∑ P(Oi)+∑ P(O i = P(A) +P(B) A∪B A B if E1, E2, …EK are collectively exhaustive events, their probability of their union is P(E 1E ∪..2∪E ) =1 K Since the events are collectively exhaustive, their union is the whole sample S. with the application of postulate 3, we will have the result above. Probability rules A: event; S: sample space; Abar is the A’s complement. The Complement rule: i.e., P(A) + P(A) =1 The Addition Rule: The probability of the union of two events is Conditional Probability Suppose that we are concerned about the probability of A, given that B has occurred The conditional probability of Event A conditional on even B, Dented as P(A|B) •P(A |B) = P(A∩B) withP(B) > 0; P(B) P(A∩B) Similarly,P(B|A) = withP(A) > 0; P(A) A Probability Tale Probabilities and joint probabilities for two events A and B are summarized in this table: Chapter 3 Probability Ben Zakaib Conditional Probability Example Multiplication rule also Statistical Independence Let A and B be two events. Theses events are said to be statistically independent if and only if From the multiplication rule it also follows that if P(B) > 0 if P(A) > 0 Chapter 3 Probability Ben Zakaib more generally, the events E1, E2, …, Ek are statistically independent if and only if: Difference difference between mutually exclusive and independent Two events are mutually exclusive if they cannot occur jointlyàthe prob. of their intersection is zero. Two events are statistically independent if the prob. of their intersection is the product of their individual prob, and in general, that prob. is not zero. If two events are mutually exclusive, then if one occurs, the other cannot, and they are not independent. Bivariate Probabilities Features The events Ai and Bk are mutually exclusive and collectively exhaustive within their sets; The intersections can occur between all events from the two sets. These intersections can be regarded as basic outcomes of a random experiment. Chapter 3 Probability Ben Zakaib The probabilities of two sets of events are called bivariate probabilities Definition of Joint and Marginal Probabilities The intersection probability are called joint probabilities; the probabilities for individual events P(Ai) or P(Bj) are called marginal probabilities. Joint and Marginal Probabilities The probability of a joint event, A ∩ B: Computing a marginal probability Where B ,1B ,2…, B ake k mutually exclusive and collectively exhaustive events Independent Events Let A and B are a pair of events, each broken into mutually exclusive and collectively exhaustive event categories denoted by labels A1, A2, …, Ah, and B1, B2, …, Bk. If every event Ai is statistically independent of every event Bj, then A and B are independent events. à “viewing frequency” and “income” are not statistically independent Odds Overinvolvement Ratio Commonly used in marketing: ad. Or promotion activities; Chapter 3 Probability Ben Zakaib The probability of event A c1nditional on event B div1ded by the probability of A 1 conditional on activity B 2s defined as the overinvolvement ratio: An overinvolvement ratio greater than 1 implies that event A incr1ases the conditional odds ratio in favor of B1: Bayes’ Theorem Let A1 and B1 be two events. Bayes’ theorem states that a way of revising conditional probabilities by using available or additional information Solution steps for Bayes’ Theorem 1. define the subset events from the problem; 2. define the probability of the events in 1; Compute the complement of the probabilities; Apply bayes’ theorem to compute the probability for the problem solution. Bayes’ theorem Example Let S = Successful well U = unsuccessful well P(S) = . 4, P(U) = .6 Define the detailed test event as D Chapter 3 Probability Ben Zakaib Conditional Probabilities: P(D|S) = .6 P(D|U) = .2 Goal to find P(S|D) P(S |D) = P(D |S)P(S) P(D| S)P(S) +P(D |U)P(U) (.6)(.4) = (.6)(.4)+ (.2)(.6) .24 = = .667 .24 +.12 Chapter 4: Discrete Random Variables and Probability distribution Ben  Zakaib Chapter 4 Interpret the mean and standard deviation for a discrete random variable Use the binomial probability distribution to find probabilities Describe when to apply the binomial distribution Use the hypergeometric and Poisson discrete probability distributions to find probabilities Explain covariance and correlation for jointly distributed discrete random variables Random Variable Random Variable – represents a possible numerical value from a random experiment X: random variable (discrete randome variable); Possible values x=1. x=2, …, x=5; each with probability 0.2 Discrete Random Variables – can take on no more than a countable number of values Continuous random variable – a random variable is a continuous random variable if it can take any value in an interval Probability Distributions for Discrete Random Variables The Probability distribution function – P(x), of a discrete random variable X expresses the probability that X takes the value x, as a function of x. That is P(x) = P(X = x), for all value of x The probability distribution function of a random variable is a representation of the probabilities of all the possible outcomes Chapter 4: Discrete Random Variables and Probability distribution Ben  Zakaib Probability Distribution required Properties Let x be a discrete random variable with probability distribution function P(x): First, 0 ≤ P(x) ≤ 1 for any value of x Second the individual probabilities sum to 1 The events X=x, for all possible value of x, are mutually exclusive and collectively exhaustive Cumulative Probability Function The cumulative probability function, denoted F(x ),0shows the probability that X does not exceed the value x 0 Derived Relationship The derived relationship between the probability distribution and the cumulative probability distribution Derived Properties Derived properties of cumulative probability distributions for discrete random variables Let X be a discrete random variable with cumulative probability distribution F(x0
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