QUANTITATIVE METHODS 1

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• QUANTITATIVE METHODS 1

SAMIR K. SRIVASTAVA
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Binomial Distribution

• Bernoulli Trial
• An experiment that has only two possible outcomes
  (Success/Failure)
• e.g. tossing a coin
• P(S) p, P(F) 1-p
• Probability p is unchanged from one trial to
  next.
• All trials are independent of each other.
• Suppose that n such trial are conducted.
• What is the number of successes in n trials?
• It is a random variable. Range 0 n.
• What is the probability distribution of this
  random variable?

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Binomial Distribution

• Consider a specific sequence of Successes and
  Failures FSSFF
• There are 2 Successes? X 2
• What is the probability of this specific
  sequence?
• (1-p)pp(1-p)(1-p) p2(1-p)3
• Is this the only sequence with 2 successes?
  SSFFF, SFSFF, SFFSF, SFFFS,
• How many? (5C2). The probability of occurrence
  for each one is p2(1-p)3
• What is the total probability of getting 2
  Successes out of 5 trials?
• (5C2) p2(1-p)3

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Binomial Distribution

• P(Xx) p(xn,p) (nCx) px(1-p)n-x
• n and p are the parameters of the distribution.
• Expected Value of a Random Variable. (Mean value)
• E(X) ?x.f(x)
• For Binomial distribution E(X) ?x n.p
• Variance V(X)
• V(X) ?(x E(x))2.f(x)
• Variance of Binomial Distribution V(X) ?x2
  n.p.(1-p)
• ?x ?n.p.(1-p)
• When is the variance large?

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Binomial Distribution - Insights
Binomial Probabilities N 10, p
0.95 x p(x) 0 0.0000 1 0.0000 2 0.0000 3 0.0000 4
0.0000 5 0.0001 6 0.0009 7 0.0105 8 0.0746 9 0.315
2 10 0.5987
Binomial Probabilities N 10, p
0.05 x p(x) 0 0.5987 1 0.3152 2 0.0746 3 0.0105 4
0.0009 5 0.0001 6 0.0000 7 0.0000 8 0.0000 9 0.000
0 10 0.0000
Binomial Probabilities N 10, p
0.50 x p(x) 0 0.0010 1 0.0097 2 0.0440 3 0.1172 4
0.2051 5 0.2460 6 0.2051 7 0.1172 8 0.0440 9 0.009
7 10 0.0010
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Applications of Binomial Distribution

• The production process of a particular product
  produces 5 defective units. A customer has
  ordered a batch of 20 units. If the batch
  contains 3 or more defective units, the customer
  will reject the entire batch, and cancel the
  order.
• What is probability that the order will be
  cancelled?
• What is the maximum permissible percentage of
  defectives in the production process so that the
  probability of rejection is reduced to less than
  1?

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Applications of Binomial Distribution

• P(Rejection) 1- P(X ?2) 1- P(X0)P(X1)P(X
  2)
• P(X ?2) (20 C 0) p0(1-p)20 (20 C 1) p1(1-p)19
  (20 C 2) p2(1-p)18 p0.05 here
• 0.35850.37740.1887 0.9246
• P(rejection) 0.0754
• Binomial Tables
• Given p, n and x, read off the probability value
  from the table
• Cumulative Probability P(X ? x)
• For what value of p, P(X ? 2) exceeds 99?
• p 0.02, P(X ? 2) 0.9929
• P 0.03, P(X ? 2) 0.9790

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Another Example

• A tyre wholesaler has 1000 Excel tyres in stock
  with 100 slightly damaged tyres randomly mixed in
  it. A retailer buys 10 tyres from this stock.
  Find the probability that he receives 8 undamaged
  tyres.
• We can use binomial here as sample size (10) is
  much lower than the size of the population (1000)
• So, n10, p0.9, (1-p) 0.1, r8
• Now, P(r) nCrpr(1-p)n-r 10C80.980.12 0.194

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Practice Problems

• If 60 of voters in a constituency prefer one
  particular candidate, what is the probability
  that in a sample of 12 voters exactly 7 will
  prefer him.
• A buyer checks large lots of batteries by
  inspecting a sample of 10 batteries and
  classifying each inspected battery as good or
  defective. She rejects the whole lot and sends it
  back to the supplier if the sample contains more
  than two defectives. Lots which are not rejected
  are accepted.
• A If 5 of batteries in the lot are defective,
  what is the probability that the lot will be
  accepted?
• B If the lot has 25 defective, what is the
  chance that it will be accepted?
• 0.9855, 0.5256

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Negative Binomial Distribution

• Consider a sequence of Bernoulli trials.
• Binomial X - No. of successes in n trials.
• Negative Binomial (also called Pascal)
• Suppose you wish to continue conducting trials
  until a desired number of successes are achieved.
• Random Variable X How many failures take place
  before the desired number, k, of successes occur?
• If k 3, and the sequence FFSFSFFFS results,
  then X 6.

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Negative Binomial Distribution

• Last trial must be a success. Why?
• xk-1 trials are conducted before the last one.
• What is the number of possible sequences with x
  failure and k-1 successes?
• xk-1Cx.
• Each one of these has probability pk-1(1-p)x
• P(Xx) (Probability of k successes in xk-1
  trials) X p
• P(xk,p) xk-1Cxpk(1-p)x
• E (X) k/p
• V (X) k(1-p)/p2

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Negative Binomial Distribution
Geometric Distribution A special case of
Negative Binomial Dist. Let k 1 How many
failures before the first success? P(xp)
p(1-p)x
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Poisson Distribution

• Consider n ? ? and p ? 0
• But Lim n.p ?
• There are 60000 vehicles on the streets of
  Bhubaneswar.
• On a given day, probability of a vehicle meeting
  with an accident is 0.00005.
• n.p ? 3 accidents per day (accident rate)
• The actual values of n and p are not important,
  as long as ? is known.
• Poisson Distribution is well-suited for such
  situations.

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Poisson Distribution

• A Bank has 8000 customers. The probability of
  customer arriving on a given day is 0.005.
• ? 40. Customer arrive at a rate of 40 per day
  (arrival rate).
• The actual number of customers arriving may range
  from 0 to ? (actually 8000).
• A machine produces 150000 parts per day, with
  0.001 probability of a part being defective. What
  is the number of defective parts produced in a
  day?
• Random variable X No. of customer arriving in
  one hour.
• No. of accidents taking place in a
  day.
• No. of defective parts produced in
  a day.
• What is the probability distribution of X?

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Poisson Distribution
E(X) ? , V(X) ?
n 10 p 0.2 0.1074 0.2684 0.3020 0.2013 0.0881
0.0264 0.0055 0.0008 0.0001 0.0000
n20 p 0.1 0.1216 0.2702 0.2852 0.1901 0.0898 0.
0319 0.0089 0.0020 0.0004 0.0001
n 40 p 0.05 0.1285 0.2706 0.2777 0.1851 0.0901
0.0342 0.0105 0.0027 0.0006 0.0001
n 100 p 0.02 0.1326 0.2707 0.2734 0.1823 0.090
2 0.0353 0.0114 0.0031 0.0007 0.0002
Poisson Tables
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Hypergeometric Distribution

• Binomial
• Select sample of size n from a potentially
  infinite population.
• View selection of n items as a sequential
  process.
• Probability p is unaffected by previous
  selections.
• What if the population is finite?
• In a population of 10 items, 5 are defective.
• p 0.5 for the first item.
• For second item, p depends on the outcome of
  first selection.
• First item defective ? p 4/9
• Not defective ? p 5/9

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Hypergeometric Distribution

• Given a population of N items containing kNp
  defectives, if n items are selected at random,
  what is the probability of getting x defective
  items? Hypergeometric Distribution.
• Range of x is from 0 to min(n,k)

• Red/Black Balls.
• N total number of balls
• Np No. of black balls (defective items)
• N-Np No. of red balls (good items)

As N becomes large, or n/N becomes small, these
probabilities tend to Binomial probabilities.
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Continuous Probability Distributions

• Uniform Distribution (Rectangular Distribution)
• Random Variable X lies in the interval (a,b),
  i.e. aXb
• All values of X are equally likely. (Uniform)
• Probability Density is the same everywhere.

1/(b-a)
P(xt)
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Expectation and Variance

• E(X) ? ?ab x.f(x).dx
• V(X) ?2 ?ab (x-?)2.f(x).dx
• For Uniform Distribution
• E(X) (ab)/2
• V(X) (b-a)2/12
• A special case a 0, b 1 Uniform(0,1)
• E(X) 0.5, V(X) 1/12

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Digression to Moments and Kurtosis

• E(X) ?1 ?-?? x.f(x).dx
• V(X) ?2 ?-?? (x-?)2 .f(x).dx
• ?3 ?-?? (x-?)3 .f(x).dx
• ?4 ?-?? (x-?)4 .f(x).dx

First Moment about zero (Mean)
Second central Moment (Variance)
Third Central Moment (Skewness)
Fourth Central Moment (Kurtosis)
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Digression to Moments and Kurtosis

• First moment defines the location
• Higher Moments define the shape
• Relative Values are more meaningful.
• Coefficient of Variation ?/?1
• Relative Kurtosis ?4 ?4/?22

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Relevance of Kurtosis
Leptokurtic ?4 gt3
Mesokurtic (Normal Dist.) ?4 3
Platykurtic ?4 lt 3
Kurtosis indicates the Peakedness of the
distribution. Normal Distribution is viewed as a
reference point, neither very high, nor very low
in terms of peakedness. Leptokurtic Higher
Peakedness compared to Normal. Platykurtic
Lower Peakedenss compared to Normal. Uniform
Distribution? Leptokurtic or Platykurtic? ?4 1.8
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Normal Distribution

• Also called Gaussian Distribution.
• Most important and most widely used.
• Symmetrical, bell shaped, extends infinitely in
  both directions.
• Many naturally occurring data follows Normal
  Distribution
• Temperature, rainfall, measurements of living
  organisms.
• Measurements of manufactured parts, errors and
  deviations from norms.

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Normal Distribution

• Density Function

• Two parameters ? - mean, ? - standard deviation

?
?2
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Normal Distribution
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Computing Normal Probabilities

• How to find P(a?x?b)?
• Evaluate the integral of f(x) from a to b.
• Alternatively, P(a?x?b) F(b) F(a)

• No closed form solution to integral of f(x).
• Numerical integration by computer is only way.
• Tables of F(x) available in text books are
  useful.
• What about the parameters ? and ??
• Do we need a separate table for each ? and ?
  combination?
• One table is enough for all. How?

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Standard Normal Distribution

• Suppose X N(?,?)
• Consider Z X- ? as a random variable.
• What is the mean and variance of Z? What is the
  distribution of Z?
• Z N(0,?)
• P(X ? a) ?
• Substitute X Z?
• P(X ? a) P(Z? ? a) P(Z ? a-?)

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Standard Normal Distribution

• X N(0,?)
• Consider Z X/? as a random variable.
• What is the mean and variance of Z? What is the
  distribution of Z?
• Z N(0,1)
• P(X ? a) ?
• Substitute X Z?
• P(X ? a) P(Z? ? a) P(Z ? a/?)

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Standard Normal Distribution

• Given XN(?,?), define Z (X- ?)/?.
• Then Z N(0,1)
• P(X ? a) P(Z ?(a- ?)/?)
• Table of F(z) P(Z ?z) are available and can be
  used to find cumulative probability for Normal
  distribution with any mean and standard
  deviation.

N(0,1)
N(?,?)
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Example

• During a socio economic survey, the concerned
  authorities came to the conclusion that the mean
  level of per capita income in the area was Rs
  1600 per month with a standard deviation of Rs
  200. The total population of the area was 400000.
• The government asked the authorities to give the
  number of people who fell into following
  categories
• Monthly per capita Income lt 1000
• 1200 lt Monthly per capita Income ? 1800
• Monthly per capita Income gt 2000

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Example
Here, X Monthly per capita Income X N (?
1600, ? 200)
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Relationship between Normal and Binomial

• Let X Binomial(n,p)
• ?x np, ?x ?np(1-p)
• Define random variable Y (X-np)/?n.p(1-p)
• As n??, distribution of Y approaches N(0,1).
• Though X is discrete, as n increases, it
  approaches continuity.
• Normal is a reasonable approximation of Binomial
  when
• np gt 5 if p lt ½
• n(1-p) gt 5 if p gt ½

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Relationship between Normal and Binomial

• Example What is the probability of 3 or fewer
  heads in 10 tosses?
• n 10, p 0.5, np ? 5, ? ?2.5 1.58
• Z3.5 (3.5-5)/1.58 -0.95
• FZ (-0.95) 0.1711
• From Binomial Tables, P(X lt 3) 0.1719

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Exponential Distribution

• Also called Negative Exponential
• f(x?) (?)exp(-?x) x ? 0
• E(X) 1/?, V(X) 1/?2, i.e. ? 1/?
• ?3 2 (Skewed Right)
• ?4 9 (Highly Leptokurtic)

F(x?) 1-exp(-?x)
? is called Arrival Rate or Failure Rate
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Memory-less Property

• Exponential distribution is used to model
  customer arrivals in queuing systems (e.g. a bank
  window)
• What is the probability that a customer will
  arrive in next 5 minutes? P(X ? 5).
• Suppose that no customer arrives in the 5
  minutes.
• Given this, what is the probability that a
  customer will arrive in next 5 minutes? P(X ? 10/
  X ? 5)
• After 10 minutes? P(X ? 15/ X ? 10)
• For exponential distribution, it is the same as
  P(X ? 5), no matter how long you have waited.

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Memory-less Property Proof

• P(X? t? / X? t) P(X ? ?) for any value of t
• A X ? t?
• B X? t
• P(A) P(X ? t?) 1-e -?(t?)
• P(B) P(X ? t) e -?t
• P(A/B) P(A?B)/P(B)
• P(A?B) P(t?X?t?) F(t?)-F(t)
• (1-e-?(t?))(1-e-?t ) e-?t - e-?(t?)
• P(A/B) 1 - e -?? P(X ? ?)

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Example

• The distribution of total time a light bulb will
  burn from the moment it is first put into service
  is known to be exponential with the mean time of
  failure between the bulbs equal to 1000 hours.
  What is the probability that the bulb will last
  more than 1000 hours?
• Here, ? (1/1000)
• And f(t) (1/1000)e-t/1000 for t ? 0 0
  otherwise.
• Therefore, P(tgt1000) 1- p(t?1000) 1-(1-
  e-1000/1000)
• e-1 0.368

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Relationship between Poisson and Exponential

• Consider Arrival of customers in bank.
• Time between two successive arrivals follows
  Exponential Dist.
• Consider an interval of time of duration t.
• How many customers will arrive in this interval?
• Discrete Random Variable.
• Follows Poisson distribution with parameter ?t.

Poisson Process
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Probability Distributions - Overview
Exponential(?)
Binomial(n,p)
Poisson(?)
Bernoulli Trial
Negative Binomial(k,p)
Normal(?,?)
Geometric(p)
Uniform(a,b)
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Central Limit Theorem

• Consider random variables X1,X2,X3........,Xn
  following any arbitrary distributions.
• Let Y X1X2X3........Xn
• What is the distribution of Y?
• As n ? ?, distribution of Y approaches Normal
  Distribution.
• Central Limit Theorem.

PLAY WITH THE SOFTWARE PROVIDED TO APPRECIATE CLT
BETTER
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Assignment 3
A supplier of machine parts has got an order to
supply piston rods to a big auto manufacturer.
The client has specified that the rod diameter
should lie between 2.541 and 2.548 cms.
Accordingly, the supplier has been looking for
the right kind of machine. He has identified two
machines, both of which can produce a mean
diameter of 2.545 cms. These machines too are not
perfect. The standard deviations of the diameters
produced from machine 1 and machine 2 are0.003
and 0.005 cm. respectively, i.e. machine 1 is
better than machine 2. This is reflected in the
prices of the machines. Machine 1 costs Rs 3.3
lacs more than machine 2. The supplier is
confident of making a profit of rs 100 per piston
rod however, a rod reject will mean a loss of Rs
40. The supplier wants to know whether he should
go for the better machine at extra cost. What is
the minimum production level that supports this
decision? Assume that there is enough demand for
piston rods.
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Thank You !