probability and statistics for management and engineering basics

8/10/2019 Probability and Statistics for Management And Engineering Basics

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Probability and Probability Distribution

Probability is the measure of the likeliness that anevent will occur.Probabilityis used to quantify an attitude of mind towards some proposition of whose

truth we are not certain. The proposition of interest is usually of the form "Will

a specificevent occur?" The attitude of mind is of the form "How certain are we

that the event will occur?" The certainty we adopt can be described in terms of

a numerical measure and this number, between 0 and 1 (where 0 indicates

impossibility and 1 indicates certainty), we call probability . Thus the higher

the probability of an event, the more certain we are that the event will occur. A

simple example would be the toss of a fair coin. Since the 2 outcomes are

deemed equiprobable, the probability of "heads" equals the probability of

"tails" and each probability is 1/2 or equivalently a 50% chance of either

"heads" or "tails".

History of Probability

"A gambler's dispute in 1654 led to the creation of a mathematical theory of

probability by two famous French mathematicians, Blaise Pascal and Pierre de

Fermat. Antoine Gombaud, Chevalier de Mr, a French nobleman with an

interest in gaming and gambling questions, called Pascal's attention to an

apparent contradiction concerning a popular dice game. The game consisted in

throwing a pair of dice 24 times; the problem was to decide whether or not to

bet even money on the occurrence of at least one "double six" during the 24

throws. A seemingly well-established gambling rule led de Mr to believe that

betting on a double six in 24 throws would be profitable, but his own

calculations indicated just the opposite.

This problem and others posed by de Mr led to an exchange of letters

between Pascal and Fermat in which the fundamental principles of probability

theory were formulated for the first time. Although a few special problems on

games of chance had been solved by some Italian mathematicians in the 15th

and 16th centuries, no general theory was developed before this famous

correspondence.

The Dutch scientist Christian Huygens, a teacher of Leibniz, learned of this

correspondence and shortly thereafter (in 1657) published the first book on
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probability; entitled De Ratiociniis in Ludo Aleae, it was a treatise on problems

associated with gambling. Because of the inherent appeal of games of chance,

probability theory soon became popular, and the subject developed rapidly

during the 18th century. The major contributors during this period were Jakob

Bernoulli (1654-1705) and Abraham de Moivre (1667-1754).

In 1812 Pierre de Laplace (1749-1827) introduced a host of new ideas and

mathematical techniques in his book, Thorie Analytique des Probabilits.

Before Laplace, probability theory was solely concerned with developing a

mathematical analysis of games of chance. Laplace applied probabilistic ideas

to many scientific and practical problems. The theory of errors, actuarial

mathematics, and statistical mechanics are examples of some of the important

applications of probability theory developed in the l9th century.

Like so many other branches of mathematics, the development of probability

theory has been stimulated by the variety of its applications. Conversely, each

advance in the theory has enlarged the scope of its influence. Mathematical

statistics is one important branch of applied probability; other applications

occur in such widely different fields as genetics, psychology, economics, and

engineering. Many workers have contributed to the theory since Laplace's time;

among the most important are Chebyshev, Markov, von Mises, and Kolmogorov.

One of the difficulties in developing a mathematical theory of probability has

been to arrive at a definition of probability that is precise enough for use in

mathematics, yet comprehensive enough to be applicable to a wide range of

phenomena. The search for a widely acceptable definition took nearly three

centuries and was marked by much controversy. The matter was finally

resolved in the 20th century by treating probability theory on an axiomatic

basis. In 1933 a monograph by a Russian mathematician A. Kolmogorov

outlined an axiomatic approach that forms the basis for the modern theory.

(Kolmogorov's monograph is available in English translation as Foundations of

Probability Theory, Chelsea, New York, 1950.) Since then the ideas have been

refined somewhat and probability theory is now part of a more general

discipline known as measure theory."

Basic Terminology

Trial

Can refer to each individual repetition when talking about an experiment

composed of any fixed number of them. As an example, one can think of an
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experiment being any number from one to n coin tosses, say 17. In this case,

one toss can be called a trial to avoid confusion, since the whole experiment is

composed of 17 ones.

Sample space

The set of possible outcomes of an experiment. For example, the sample space

for rolling a six-sided die will be {1, 2, 3, 4, 5, 6}

Event

A subset of the sample space (a possible experiment's outcome), to which a

probability can be assigned. For example, on rolling a die, "getting a five or a

six" is an event (with a probability of one third if the die is fair)

Compound Event

A compound event consists of two or more simple events. Tossing two dice is a

compound event.

Exhaustive Events

The total possible outcomes of a trail.For example in In a throw of a

die,Number of exhaustive events = 6

Favorable Events:The outcomes of a trail which cause the happening of a particular event.For

example in a event A = Getting an even number = {2, 4, 6},Number of favorable

events = 3

Equally likely Events:

The events are said to be equally likely events, if none of them is expected to

occur in preference to other.For example In a toss of an unbiased coin,P (H) = P

(T) =

Mutually Exclusive/ Disjoint Event

The events which can not occur simultaneously.For example In a draw of a card

from a deck of playing car ds if A = The card drawn is a club B = The card drawn
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is a heart then Events A and B are mutually exclusive events.

Complementary Event()

The non-happening of event E is called complementary event EC of event E.

P() = 1 P(E).For example if probablity of getting a head is 0.4 thenprobabilty of tail is 0.6.

Independent Event

The happening/non-happening of one event does not depend on the occurrence

of other event

Dependent Event

The events which are not independent events.

Fomulae for Probablitity

If a random experiment results in n exhaustive, mutually exclusive and equally

likely events, out of which m are favorable to the happening of event E, then

the probability of occurrence of event E is

Probability can be expressed in terms of fraction, percentage, decimal or ratio.

Facts about probability

Probability of each event is a number between 0 and 1 incl usive i. e., 0

P(E) 1

Probability of impossible event is zero.

[Note: the converse is not necessarily true]


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Probability of certain event is one.

The sum of probabilities of all possible events is equals to one i.e., P(E)

=1

Approaches of Probability

Classical theory of probability

The classical approach to probability is to count the number of favorable

outcomes, the number of total outcomes (outcomes are assumed to be mutually

exclusive and equiprobable), and express the probability as a ratio of these two

numbers. Here, "favorable" refers not to any subjective value given to the

outcomes, but is rather the classical terminology used to indicate that anoutcome belongs to a given event of interest. What is meant by this will be

made clear by an example, and formalized with the introduction of axiomatic

probability theory.

Classical definition of probability

If the number of outcomes belonging to an event is ,

and the total number of outcomes is , then the

probability of event is defined as .

Empirical or Statistical Probability or Frequency of

occurrence

This approach to probability is well-suited to a wide range of scientific

disciplines. It is based on the idea that the underlying probability of an event

can be measured by repeated trials.


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Empirical or Statistical Probability as a measure of

frequency

Let be the number of times event occurs after

trials. We define the probability of event as

It is of course impossible to conduct an infinite number of trials. However, it

usually suffices to conduct a large number of trials, where the standard of large

depends on the probability being measured and how accurate a measurement

we need.

Axiomatic probability theory

Axiomatic probability theory, although it is often frightening to beginners, is

the most general approach to probability, and has been employed in tackling

some of the more difficult problems in probability. We start with a set of

axioms, which serve to define a probability space. Although these axioms may

not be immediately intuitive, be assured that the development is guided by the

more familiar classical probability theory. Let S be the sample space of a

random experiment. The probability P is a real valued function whose domain is

the power set of S and range is the interval [0,1] satisfying the following

axioms:

(i) For any event E, P (E ) 0

(ii) P (S) = 1

(iii) If E and F are mutually exclusive events, then P(E F) = P(E) + P(F).

It follows from (iii) that P() = 0. To prove this, we take F = and note that E

and are disjoint events. Therefore, from axiom (iii), we get P (E ) = P (E) +P () or P(E) = P(E) + P () i.e. P () = 0. Let S be a sample space containing

outcomes 1 , 2 ,...,n , i.e., S = {1, 2, ... , n}

It follows from the axiomatic definition of probability that:

(i) 0 P (i) 1 for each i S


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(ii) P (1) + P (2) + ... + P (n) = 1

(iii) For any event A, P(A) = P (i ), i A .

Conditional Probability P(A|B) : The probability of event A provided

event B has already happened.

P (A|B) =

Concept

If an event B has occurred, instead of S, we consider B only.

The conditional probability of A given B will be the ratio of that part of A which

is included in B i.e. P(AB) to the probability of B.

Addition Theorem:For two events A and B, probability of happeningatleast one of them is

P(AB) = P(A) + P(B) P (AB)

Multiplication Theorem : For two events A and B, probability of theirsimultaneous happening is P(A B) = P(A) * P(B|A), P(A) > 0

Or P(A B) = P(B) *P(A|B), P(B) > 0

If the events A and B are independent i.e. P(A|B) = P(A) & P(B|A) = P(B), then

P(A B) = P(A) *P(B)

Bayes Theorem (aka, Bayes Rule)

Bayes' theoremprovides a mathematicalrule for revising anestimate or

forecast in light of experience and observation. It differs from other methods of

hypothesis testing in that itassigns 'after thefact'(posterior)probabilities to

the hypotheses instead of just accepting or rejecting them. Named after its

proponent, the UK mathematician Thomas Bayes (1702-1761) who researched

probability andstatistical inference

)(

)(

BP

BAP
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Applications of Bayes theorem

Bayes's theorem is significantly important in inverse problem theory, where the

a posterioriprobability density function is obtained from the product of priorprobability density function and the likelihood probability density function. An

important application is constructing computational models of oil reservoirs

given the observed data.

Although Bayes's theorem is commonly used to determine the probability of an

event occurring, it can also be applied to verify someone's credibility as a

prognosticator. Many pundits claim to be able to predict the outcome of an

event; political elections, trials, the weather and even sporting events. Larry

Sabato,founder ofSabato's Crystal Ball, is a perfect example. His website

provides free political analysis and election predictions. His success at

predictions has even led him to be called a "pundit with an opinion for every

reporter's phone call." We even havePunxsutawney Phil,the famous

groundhog, who tells us whether or not we can expect a longer winter or an

early spring. Bayes's theorem tells us the difference between who's on a hot

streak and who is what they claim to be.

Let's say we live in an area where everyone gambles on the outcome of coin

flips. Because of that, there is a big business for predicting coin flips. Suppose

that 5% of predictors can actually win in the long run, and 80% of those are

winners over a 2-year period. 95% of predictors are pretenders who are just

guessing, and 20% of them are winners over a 2-year period (everyone gets

lucky once in a while). This means that 82.6% of bettors who are winners over

a 2-year period are actually long-term losers who are winning above their real

average.

Probability Distribution

Random Variables

Introduction

A random variable associates each outcome of a random experiment with a

probability and it represents a measurable aspect or property of the outcomes.
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For these reasons, a random variable is essential to be defined to assume a real

number to each and every outcome of the random experiment.

Random Variable

Let S be a sample space of a random experiment. A random variable is afunction that associates a real number with each element in the sample space, S.

Types of Random Variables

Random variables are commonly classified into two types namely, discrete

random variables and continuous random variables.

Discrete Random Variable

A random variable on a random sample S is said to be discrete if the range of X

is countable. In other words, a discrete random variable is a random variable

that can assume at most a finite or a countably infinite number of possible

values.

Probability Function (or) Probability Mass Function of a

Discrete Random Variable

Let X be a discrete random variable on a sample space S, which can take the

values

,1x ,2x

nx

such thatiii pxPxXP )()(

(called the probability ofix

).The function i

pis called the probability mass function (p.m.f) of the random

variable X i f ip

must satisfy the following conditions: (i)0ip for all i and

(ii)

n

iip 1


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Distribution Function of the Discrete Random Variable

The distribution function or briefly the cumulative distribution function (c.d.f)

for a discrete random variable X is defined by

x

iX xXPxXPxFxF )()()()(

where x is any real number, that is, x .

Continuous random variable

A random variable X is said to be continuous if it takes all possible values

between certain limits o r in an interval which may be finite or infinite.

Probability Density function of a Continuous Random Variable

If X is a continuous random variable such that dxxf

dx

xX

dx

xP )(}22{ ,

then )(xf is called the probability density function (p.d.f) of X provided )(xf

satisfies the following conditions:

(i) 0)( xf , for all Rx ; (ii)

1)( dxxf

and (iii)

b

a

dxxfbxaP )()(

.

Distribution Function of the Continuous Random Variable

The cumulative distribution function (c.d.f.) or briefly the distribution function

for a continuous random variable X is defined by

x

X dxxfxXPxFxF )()()()(

, . x


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Relationship between probability distribution function )(xF and probability

density function )(xf :

From the definition of probability distribution function, we can conclude

that)())(( xfxF

dx

d

, that is, the probability density function of a continuous

random variable X , )(xf is the derivative of the probability distribution

function of the random variable, )(xF

Binomial distribution

Frequency distribution where only two (mutually exclusive)outcomes are

possible, such as better or worse, gain or loss, head ortail,rise or fall,success

or failure,yes or no. Therefore, if the probability of success in any given trial is

known, binomial distributions can beemployed to compute a given number of

successes in a given number of trials.And it can be determined if anempirical

distribution deviates significantly from a typical outcome. Alsocalled Bernoullidistribution after its discoverer, the Swiss mathematician Jacques Bernoulli

(1654-1705)

A binomial experiment is one that possesses the following properties:

The experiment consists of n repeated trials;

Each trial results in an outcome that may be classified as a success or a failure

(hence the name, binomial);

The probability of a success, denoted by p, remains constant from trial to trialand repeated trials are independent.

The number of successes X in n trials of a binomial experiment is called a

binomial random variable.

The probability distribution of the random variable X is called a binomial

distribution, and is given by the formula:
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where

n = the number of trials

x = 0, 1, 2, .. . n

p = the probability of success in a single trial

q = the probability of failure in a single trial

(i.e. q = 1 p)

P(X) gives the probability of successes in n binomial trials.

Mean and Variance of Binomial Distribution

If p is the probability of success and q is the probability of failure in a binomial

trial, then the expected number of successes in n trials (i.e. the mean value of

the binomial distribution) is

E(X) = = np

The variance of the binomial distribution is

V(X) = 2 = npq

Note: In a binomial distribution, only 2 parameters, namely n and p, are needed

to determine the probability.

Poisson Distribution

The Poisson Distribution was developed by the French mathematician Simeon

Denis Poisson in 1837.

The Poisson random variable satisfies the following conditions:


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The number of successes in two disjoint time intervals is independent.

The probability of a success during a small time interval is proportional

to the entire length of the time interval.

Apart from disjoint time intervals, the Poisson random variable alsoapplies to disjoint regions of space.

Applications

the number of deaths by horse kicking in the P russian army (first

application)

birth defects and genetic mutations

rare diseases (like Leukemia, but not AIDS because it is infectious and so

not independent) - especially in legal cases

car accidents

traffic flow and ideal gap distance

number of typing errors on a page

spread of an endangered animal in Africa

failure of a machine in one month

The probability distribution of a Poisson random variable X representing the

number of successes occurring in a given time interval or a specified region of

space is given by the formula:

Mean and Variance of Poisson Distribution

If is the average number of successes occurring in a given time interval or

region in the Poisson distribution, then the mean and the variance of the

Poisson distribution are both equal to .


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E(X) = and V(X) = 2 =

Note: In a Poisson distribution, only one parameter, is needed to determine

the probability of an event

Normal Distibrution

The Normal Probability Distribution is very common in the field of statistics.

Whenever you measure things like people's height, weight, salary, opinions or

votes, the graph of the results is very often a normal curve.

The Normal Distribution

A random variable X whose distribution has the shape of a normal curve iscalled a normal random variable.

Normal Curve

This random variable X is said to be normally distributed with mean and

standard deviation if its probability dist ribution is given by


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Properties of a Normal Distribution

The normal curve is symmetrical about the mean ;

The mean is at the middle and divides the area into halves;

The total area under the curve is equal to 1;

It is completely determined by its mean and standar d deviation (or

variance 2)

Note: In a normal distribution, on ly 2 parameters are needed, namely and 2.

Area Under the Normal Curve using Integration

The probability of a continuous normal variable X found in a particular interval

[a, b] is the area under the curve bounded by x=a and x=b and is given by

P(a


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We can transform all the observations of any normal random variable X with

mean and variance to a new set o f observations of another normal random

variable Z with mean 0 and variance 1 using the following transformation:

Z=

A standard normal distribution is a normal distribution with mean 0 and

standard deviation 1. Areas under this curve can be found using a standard

normal table.

Standard Normal table


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probability and statistics for management and engineering basics

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