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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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