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Binomial and Normaldistributions used in
business forecastingMade By:
Abhay Singh
Roll No.50202
BBS I A
Business Statistics
and Applications
Term Paper
By Abhay Singh
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STATISTICS TERM PAPER
Figure
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What are theoretical distributions?Theoretical distributions are a more scientific way of drawing inferences about
the population characteristics. In the population, the value of the variable may
be distributed according to some definite probability law which can be expressed
mathematically and the corresponding probability distribution is known as the
theoretical probability distribution.
Such probability laws may be based on a prior considerations or a posteriori
inferences. These distributions are based on expectations on the bass of previous
experience. Theoretical distributions also enable us to fit a mathematical model
or a function of the form y = p(x) to the given
data.
Refer to Figure on page 4
Figure
Binomial DistributionIn probability theory and statistics, the binomial distribution is the discrete
probability distribution of the number of successes in a sequence
ofnindependent yes/no experiments, each of which yields success
with probabilityp. Such a success/failure experiment is also called a Bernoulli
experiment or Bernoulli trial; when n = 1, the binomial distribution is a Bernoulli
distribution. The binomial distribution is the basis for the popular binomial
test ofstatistical significance.
The binomial distribution is frequently used to
model the number of successes in a sample of
size n drawn with replacement from a
population of size N. If the sampling is carried out
without replacement, the draws are not
independent and so the resulting distribution is a
hypergeometric distribution, not a binomial one. However, for N much larger
than n, the binomial distribution is a good approximation, and widely used.
DiscreteProbabilit
Continuous
Probabilit
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Cumulative Distribution Function
Probability Mass Function
Why is it important?
The binomial distribution is widely used to test statistical probabilities and
significance, and is a good way of visually detecting unexpected values. It is a
useful tool in determining permutations, combinations, and probabilities, wherethe outcomes can be broken Binomial Distribution into two probabilities
(p and q), where p and q are complementary (i.e., p + q = 1).
For example, tossing a coin has only two possible outcomes, heads or tails. Each
of these outcomes has a theoretical probability of 0.5. Using the binomial
expansion, showing all possible outcomes and combinations, the probability is
represented as follows:
(p + q)2 = p2 + 2pq +q2, or more simply, pp + 2pq + qqIf p is heads and q is tails, the theory shows there is only one way to get two
heads (pp), two ways to get a head and a tail (2pq), and one way to get two tails
(qq).
Common uses of binomial distributions in business include quality control, public
opinion surveys, medical research, and insurance problems. It can be applied to
complex processes such as sampling items in factory production lines or to
estimate percentage failure rates of products and components.
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Quick facts
To satisfy the requirements of binomial distribution, the event being studied must
display certain characteristics:
the number of trials or occurrences are fixed
there are only two possible outcomes (heads/tails or win/lose, for example)
all occurrences are independent of each other (tossing a head does not
make it more or less likely you will get the same result next time)
all outcomes have the same probability of success
Binomial distribution is best applied in cases where the population size is
at least 10 times the sample size, and not to simple random samples.
To find probabilities from a binomial distribution, you can perform amanual calculation, but there are online calculators available, or you can
use a binomial table or computer spreadsheet.
The binomial distribution is sometimes called a Bernoulli experiment or
trial.
The binomial probability refers to the probability that a binomial
experiment results in exactly x successes. In example above, we see that
the binomial probability of getting exactly one head in two coin flips is 0.5.
A cumulative binomial probability refers to the probability that the
binomial random variable falls within a specified range (for example, is
greater than or equal to a stated lower limit and less than or equal to a
stated upper limit).
Uses in Business
1. Quality Control
In statistical quality control, the p-chart is a type ofcontrol chart used to monitor the proportionofnonconforming unitsin a sample, where the sample proportion nonconforming is defined as the
ratio of the number of nonconforming units to the sample size, n.
The p-chart only accommodates "pass"/"fail"-type inspection as determined by one or more go-no go
gauges or tests, effectively applying the specifications to the data before they are plotted on the chart.
Other types of control charts display the magnitude of the quality characteristic under study, making
troubleshooting possible directly from those charts.
The binomial distributionis the basis for the p-chart and requires the following assumptions [2]:267:
The probability of nonconformityp is the same for each unit;
Each unit is independent of its predecessors or successors;
The inspection procedure is same for each sample and is carried out consistently fromsample to sample
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A P-Chart
2. Public Opinion SurveyWhen doing a public opinion poll before an election in a country, they usually
approximate the hypergeometric distribution with a binomial distribution and
then using the normal approximation:
m +/-1.96*sqrt(m*(1-m)/n) to calculate a 95% confidence interval?
m= mean
n= number of people in the poll
is it as simple as that? assuming party A gets 30% of the votes, and 2000 voted
we get a 95% statistically significant interval of:30 +/-1.96*sqrt(0.30*(0.7)/2000)
However if we make a poll within a defined population of lets say 5000 people.
and the number of people in the poll is 3000. party A gets 30% of the votes in
the poll. What can we say about the total population of 5000.
here we should use the hypergeometric distribution and cannot use the binomial
approximation.
3. Medical Research
A binomial distribution can be used to describe the number of times an event willoccur in a group of patients, a series of clinical trials, or any other sequence of
observations. This event is a binary variable: It either occurs or it doesn't. For
example, when patients are treated with a new drug they are either cured or not;
when a coin is flipped, the result is either a head or tail. The binary outcome
associated with each event is typically referred to as either a "success" or a
"failure." In general, a binomial distribution is used to characterize the number of
successes over a series of observations (or trials), where each observation is
referred to as a "Bernoulli trial."
In a series ofn Bernoulli trials, the binomial distribution can be used to calculatethe probability of obtaining ksuccessful outcomes. If the variable X represents
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the total number of successes in n trials, it can only take on a value from 0 to n.
The binomial distribution can be used to calculate the probability of
obtaining ksuccesses in n trials is calculated as follows:
where 0 less than or equal to p less than or equal to 1 is the probability of
success, and n!= 1 2 3[.dotmath][.dotmath][.dotmath][.dotmath](n2)(n1)n.
The above formula assumes that the experiment consists ofn identical trials that
are independent from one another, and that there are only two possible
outcomes for each trial (success or failure). The probability of success (p) is also
assumed to be the same in each of the trials.
To further illustrate the application of the above formula, if a drug was developed
that cured 30 percent of all patients, and it was administered to ten patients, the
probability that exactly four patients would be cured is:
Like other distributions, the binomial distribution can be described in terms of a
mean and the spread, or variance, of values. The mean value of a binomial
random variable X (i.e., the average number of successes in n trials) can be
obtained by multiplying the number of trials byp (np). In the above example, the
average number of persons cured in any group of 10 patients would thus be 3.
The variance of a binomial distribution is np (1p). The variance is largest for
p = 0.5, while it decreases as p approaches 0 or 1. Intuitively, this makes sense,
since when p is very large or small nearly all the outcomes take on the same
value. Returning to the example, a drug that cured every patient p would equal
one, while for a drug that cured no one, p would equal zero. In contrast, if thedrug was effective in curing only half of the population (p = 0.5) it would be
more difficult to predict the outcome in any particular patient, and in this case
the variability is relatively large.
In studies of public health, the binomial distribution is used when a researcher is
interested in the occurrence of an event rather than in its magnitude. For
instance, smoking cessation interventions may choose to focus on whether a
smoker quit smoking altogether, rather than evaluate daily reductions in the
number of cigarettes smoked. The binomial distribution plays an important role
in statistics, as it is likely the most frequently used distribution to describe
discrete data.
4. Insurance Sector
There is similarity between insurance and game of chance and thereforeunderstandingthe concept of probability and its application to general insurance is ofimportance to us.When we toss a coin, we say that the probability of getting a head is .
There are two
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possibilities head or tail out of which one possibility i.e. head isfavourable andtherefore we say that probability of getting head is or 50%. This is thetheoretical wayof calculating probability called a priori i.e. prior to experience. In
contrast, we cannot deduce theoretically the probability of a car being stolen within theyear. For handlingproblems of this nature, we need to have data about the total number ofcars and theproportion that is stolen. There is another way of looking at probability ofgetting head intossing of coin which is more relevant for our purpose. If we go on tossingthe coin andnote the number of heads coming and if ideally this tossing is continued
for infinitenumber of times we will find that the proportion of head coming is .Every time we aretossing, we are in effect generating experience. The fact that probabilityinvolves longrun concept is important in the general insurance contract. Further head &tail aremutually exclusive events. The idea of mutual exclusivity can apply forexample tocalculating the probability of an injured employee being male or female,injured or killed,
damages being above or below certain level, etc. We may say that if theevent is certainto happen the probability is one and if the happening is absoluteimpossibility theprobability is zero. If the probability of happening a claim is one i.e. acertainty noinsurance company will assume such a risk except perhaps by chargingthe premiumwhich is more than the sum insured. If the happening is an impossibilityi.e. probabilityis zero, nobody would like to insure it. Between these two extremes, liesthe various riskthat come for insurance. The higher the probability of claims happening,the highershould be the premium. Probability thus attaches a numerical value to ourmeasurementof the likelihood of an event occurring. We shall now examine the law oflarge numberand the concept of probability distribution. We shall also see how theseprobabilitydistribution help us in estimating the number of claims that will be
reported in future
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during a given period of time and what will be the size of these claims.The law of largenumbers in simple terms means that the larger the data, the moreaccurate will be theestimate made. In other words the larger the sample, the more accurate
will be theestimates of the population parameters. In general insurance it wouldmean that the largerthe past data about claims, the better will be the estimate of theprediction about claimfrequency and size. It is assumed that the claim will occur in future asthey have occurredin the past. What is a probability distribution? It is the listing of all possiblevalues of arandom variable along with their associated probabilities. For our purpose,
the probabilitydistribution can be considered to be a mathematical model which candescribe the actual probability distribution. Of course, the actualprobability estimated from the availabledata will rarely coincide with those generated by the theoreticaldistribution. But the lawof large number says that it will tend closer & closer if we have sufficientlylargedatabase. Even if the data available is not extensive, we can make use ofvarioustheoretical distributions to make meaningful inferences about the behaviour of
datarelating to a particular insurance portfolio. The fact that this theoreticaldistribution canbe completely summarized by a small number of parameters is of great help.
The shapeof distribution is determined by its parameters. Parameters are numericalcharacteristicsof population. If we have set of data relating to say claims size, we cannot makebest useof them in their raw form. We may be interested to know about the average sizeof theclaim. We have a whole set of measures called the measure of central tendency.
Similarlyto properly understand the significance of the data, it is essential to know thevariabilityof data around the central tendency. In case the variance is too high, may be onehas todecide about the required reinsurance support. Yet another aspect to properlyunderstandthe given set of data is the Skew ness aspect. The distribution may be verysymmetricor it may be skewed having long trail to the right (positively skewed). Many ofthe
distribution we encounter in general insurance is skew with long tail to the right.We havea measure of this skew ness which is zero for symmetric distribution. Positive for
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positively skewed distribution and negative for negatively skewed distribution(long tailto the left). Again a knowledge and measure of flatness or peaked ness of adistributionis important for us. So we have what is called a measure of kurtosis. These
aspects ifknown properly can help in better claims management. Fortunately for us, therearetheoretical distribution models which approximate the existing claims datarelating tovarious risk categories. The actuaries make use of these models. These providemethodsof summarizing aspects of complexities. Some distributions are continuous innature andmay relate to claim size distribution and in the analysis of heterogeneity. Theothersrelate to discrete variables and hence are helpful in studying the claim numbers
distribution.
Normal Distribution
In probability theory, the normal (or Gaussian) distribution is a continuous
probability distribution that is often used as a first approximation to describe
real-valued random variables that tend to cluster around a single mean value.
The graph of the associated probability density function is "bell"-shaped, and is
known as the Gaussian function or bell curve.
where parameter is the mean or expectation (location of the peak) and 2 is
the variance, the mean of the squared deviation, (a "measure" of the width of
the distribution). is the standard deviation. The distribution with =
0 and 2 = 1 is called the standard normal.
The normal distribution is considered the most prominent probability
distribution in statistics. There are several reasons for this: First, the normal
distribution is very tractable analytically, that is, a large number of results
involving this distribution can be derived in explicit form. Second, the normal
distribution arises as the outcome of the central limit theorem, which states
that under mild conditions the sum of a large number ofrandom variables is
distributed approximately normally. Finally, the "bell" shape of the normal
distribution makes it a convenient choice for modelling a large variety of
random variables encountered in practice.
For this reason, the normal distribution is commonly encountered in practice,
and is used throughout statistics, natural sciences, and social sciences as a
simple model for complex phenomena. For example, the observational
error in an experiment is usually assumed to follow a normal distribution, and
the propagation of uncertainty is computed using this assumption. Note thata normally-distributed variable has a symmetric distribution about its mean.
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Quantities that grow exponentially, such as prices, incomes or populations,
are often skewed to the right, and hence may be better described by other
distributions, such as the log-normal distribution or Pareto distribution. In
addition, the probability of seeing a normally-distributed value that is far (i.e.
more than a few standard deviations) from the mean drops off extremely
rapidly. As a result, statistical inference using a normal distribution is not
robust to the presence ofoutliers(data that is unexpectedly far from the
mean, due to exceptional circumstances, observational error, etc.). When
outliers are expected, data may be better described using a heavy-
tailed distribution such as the Student's t-distribution.
Probability Density Function
Cumulative Descriptive
Function
Uses
1. Modern Portfolio Theory
Modern portfolio theory (MPT)or portfolio theorywas introduced
by Harry Markowitz with his paper "Portfolio Selection," which appeared
in the 1952Journal of Finance. Thirty-eight years later, he shared a Nobel
Prize with Merton Miller and William Sharpefor what has become a broad
theory for portfolio selection.
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Prior to Markowitz's work, investors focused on assessing the risks and
rewards of individualsecurities in constructing their portfolios. Standard
investment advice was to identify those securities that offered the best
opportunities for gain with the least risk and then construct a portfolio
from these. Following this advice, an investor might conclude thatrailroad stocks all offered good risk-reward characteristics and compile a
portfolio entirely from these. Intuitively, this would be foolish. Markowitz
formalized this intuition. Detailing a mathematics ofdiversification, he
proposed that investors focus on selecting portfolios based on their overall
risk-reward characteristics instead of merely compiling portfolios from
securities that each individually have attractive risk-reward
characteristics. In a nutshell, inventors should select portfolios not
individual securities.
If we treat single-period returns for various securities as random variables,
we can assign them expected values, standard
deviations and correlations. Based on these, we can calculate the
expected return and volatility of any portfolio constructed with those
securities. We may treat volatility and expected return as proxy's for risk
and reward. Out of the entire universe of possible portfolios, certain ones
will optimally balance risk and reward. These comprise what Markowitz
called an efficient frontier of portfolios. An investor should select a
portfolio that lies on the efficient frontier.
James Tobin (1958) expanded on Markowitz's work by adding a risk-free
asset to the analysis. This made it possible to leverage or deleverage
portfolios on the efficient frontier. This lead to the notions of a super-
efficient portfolio and the capital market line. Through leverage, portfolios
on the capital market line are able to outperform portfolio on the efficient
frontier.
Sharpe (1964) formalized the capital asset pricing model(CAPM). This
makes strong assumptions that lead to interesting conclusions. Not only
does the market portfolio sit on the efficient frontier, but it is actually
Tobin's super-efficient portfolio. According to CAPM, all investors should
hold the market portfolio, leveraged or de-leveraged with positions in the
risk-free asset. CAPM also introduced beta and relates an asset's expected
return to its beta.
Portfolio theory provides a broad context for understanding the
interactions ofsystematic risk and reward. It has profoundly shaped howinstitutional portfolios are managed, and motivated the use of passive
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investment management techniques. The mathematics of portfolio theory
is used extensively in financial risk management and was a theoretical
precursor for today's value-at-risk measures.
2. Human resource ManagementA disadvantage shared by all employee-comparison systems is that of employee
comparability. This has two aspects. The first has been mentioned: are the jobs
sufficiently similar? The second is whether employees are rated on the same
criteria. It is likely that one employee rates high for one reason and another rates
low for an entirely different reason. Another disadvantage is that raters do not
always have sufficient knowledge of the people being rated. Normally the
immediate supervisor has this knowledge, but in large ranking systems,
supervisors two and three levels removed often have to do the rating. The very
size of units also poses a problem. The larger the number of employees to be
ranked, the harder it is to do so; on the other hand, the larger the number in the
group, the more logical it is that there is a normal distribution. This brings up one
last problem. If the manager knows that some employees must be rated below
average, he or she will start thinking of those employees that way. This leads to a
self-fulfilling prophecy: the manager now treats them as if they cannot do well,
and they respond by not doing well.
Forecasting
The most comprehensive methodology for comparing forecast data and demand
data is to graph the cumulative results for a given period which include upper
and lower control limits (calculated from historic demand based on a normal
distribution curve). The following example shows how easy it is to quickly identify
when the demand is in control and when the demand is out of control.
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This graph shows a period of meeting the plan (Note A) and then a period when
the actual revenues start to deviate from the forecast (Note B). At this point, the
demand is still within the control parameters of "normal" demand and a
cautionary watch may be put in place, although no action is required. However, it
is clear at Note C the lower control limit has been exceeded and the "normal"
expected demand is not being met. This is the time for action and the process togain a complete understanding of the error should be invoked. Supplemental
charts will be necessary to analyze what is causing the deviation. These charts
are similar to the one shown above, but with a separate breakdown for units,
average selling price, product types, sales channels, customer and sales agent.
A possible result is finding that the graph represents a normal trend in the
business with no corrective actions necessary. It is also possible that the total
revenue versus forecast may be in sync (note A), however mismatches may exist
in the unit, average selling price, customer or product type mixtures. In each
case, this information would not be known unless this type of analysis wereavailable as well as being in place for some time in order to understand the long
term trends as well.
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Applications:
1. Using the Normal Distribution to Determine the
Lower 10% Limit of Delivery Times
A pizza deliveryman's delivery time is normally distributed with a mean of 20 minutes
and a standard deviation of 4 minutes. What delivery time will be beaten by only 10%
of all deliveries?
Problem Parameter Outline
Population Mean = = "mu" = 20 minute
Population Standard Deviation = = "sigma" = 4 minutes
x = ?
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Probability that (delivery time x) = 10% = 0.10
Delivery time is Normally distributed
Normal curve is not standardized ( 0, 1)
Problem Solving Steps
We know that Delivery Time data is Normally distributed and can therefore be
mapped on the Normal curve.
We are trying to determine Delivery Time will be lower than 90% of all Delivery
Times.
This probability corresponds to the x value at which 90% of area under the
Normal curve has a greater value and is to the right this x value. This x value
must therefore be in the left tail of the Normal curve.
If we know that 90% of the area under the Normal curve is to the right of this x
value (this is illustrated in the graph below), then we know that 40% of the total
area under the Normal curve is between this x value and the mean. The
remaining 50% of the area under the Normal curve makes up the half the
Normal curve that is on the opposite side (the right side) of the mean.
If we know that 40% of the area under the Normal curve is between this x value
and the mean, we can use the Z Score Chart to determine how many standard
deviations this x value is from the mean. The Z Score Chart belowshows this x
value to be 1.28 standard deviations from the mean.
If we know how many standard deviations this x value is from the mean, we can
use the following formula to calculate the x value, as follows:
z = ( x - ) /
x = z * +
x = (-1.28) * 4 + 20 = 14.87
The delivery time of 14.87 minutes is faster (smaller) than 90% of all delivery
times. This is illustrated in the graph below.
Answer: The delivery time of 14.87 minutes is faster (smaller) than 90% of
all delivery times.
http://excelmasterseries.com/Excel_Statistical_Master/Normal-Distribution.php#Problem%203%20-%20Graph1http://excelmasterseries.com/Excel_Statistical_Master/Normal-Distribution.php#Problem%203%20-%20Z%20ScoreCharthttp://excelmasterseries.com/Excel_Statistical_Master/Normal-Distribution.php#Problem%203%20-%20Graph1http://excelmasterseries.com/Excel_Statistical_Master/Normal-Distribution.php#Problem%203%20-%20Graph1http://excelmasterseries.com/Excel_Statistical_Master/Normal-Distribution.php#Problem%203%20-%20Z%20ScoreCharthttp://excelmasterseries.com/Excel_Statistical_Master/Normal-Distribution.php#Problem%203%20-%20Graph1 -
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2. Finding the probability of a certain type of
package passing down a conveyor belt if the
probability of that type of package passing by
is known.
Problem: A conveyor belt brings packages to a truck to loaded. The packages are
either black or white. The probablity that a package is black is 40%. What is the
probability that out of the next 10 packages, at least 2 are black and 2 are white?
The only possibility of at least two packages being white and 2 black would occur if
the number of black packages equaled 0, 1, 9, or 10. The probability of at least 2
packages being black and 2 white would therefore equal 1 minus the probability that
the number of black packages equals 0, 1, 9, or 10.
Probability of Success (Black Package) = p = 0.40
Probability of No Success (White Package) = q = 1 - p = 0.60
Number of Trials (Transactions) = n = 10
Exact Number of Successes = k = 0, 1, 9, 10
This problem uses Binomial formulas (Sample Occurrence Formulas) because
what is being measured is the mean number of occurrences from a large
number of individual samples that each have only two possible outcomes.
Probability of at least 2 Black and 2 White Packages in 10 =
1 - [ Pr(X=0) + Pr(X=1) + Pr(9) + Pr(10) ]
PR (X = 0) = f(k; n, p) = n! / [ k! * ( n - k )! ] * pk * q(n-k)
PR (X = 0) = f(0; 10, 0.40) = 0! / [ 10! * ( 10 - 0 )! ] * (0.40)\0 * (0.60)(10-0)
PR (X = 0) = 0.0060 = 0.60%
PR (X = 1) = f(k; n, p) = n! / [ k! * ( n - k )! ] * pk * q(n-k)
PR (X = 1) = f(1; 10, 0.40) = 1! / [ 10! * ( 10 - 1 )! ] * (0.40)\1 * (0.60)(10-1)
PR (X = 1) = 0.040 = 4.0%
PR (X = 9) = f(k; n, p) = n! / [ k! * ( n - k )! ] * pk * q(n-k)
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PR (X = 9) = f(9; 10, 0.40) = 9! / [ 10! * ( 10 - 9 )! ] * (0.40)\9 * (0.60)(10-9)
PR (X = 9) = 0.002 = 0.2%
PR (X = 10) = f(k; n, p) = n! / [ k! * ( n - k )! ] * pk * q(n-k)
PR (X = 10) = f(10; 10, 0.40) = 10! / [ 10! * ( 10 - 10 )! ] * (0.40)10 * (0.60)(10-10)
PR (X = 10) = 0.0 = 0.0%
1 - [ Pr(X=0) + Pr(X=1) + Pr(9) + Pr(10) ] = 1 - [ 0.006 + 0.040 + 0.002 + 0.0 ]
= 1 - 0.048 = 0.952 = 95.2%
There is a 95.2% probability at at least 2 packages will be black and 2 packages
will be white out of the next 10 packages if the probability of a package being
black is 40%.
I
Insurance, 67
T
the binomial distribution. See Page 2
Theoretical distributions, 2