Normal and Poisson Distributions

1
Normal and Poisson Distributions

• GTECH 201
• Lecture 14

2
Sampling

• Population
• The entire group of objects about which
  information is sought
• Unit
• Any individual member of the population
• Sample
• A part or a subset of the population used to gain
  information about the whole
• Sampling Frame
• The list of units from which the sample is chosen

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Simple Random Sampling

• A simple random sample of size n is a sample of n
  units chosen in such a way that every collection
  of n units from a sampling frame has the same
  chance of being chosen

4
Random Sampling in R

• In R you can simulate random draws
• For example, to pick five numbers at random from
  the set 140, you cangt sample(140,5)1 4
  30 28 40 13

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Sampling with Replacement

• Default in R is without replacementsample(c
  ("H", "T"), 10, replaceT)1 "T" "T" "T" "T"
  "T" "H" "H" "T" "H" "Tprobc(.9,.1)
• sample(c("S", "F"), 10, replaceT, prob)

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Random Number Tables

• A table of random digits is
• A list of 10 digits 0 through 9 having the
  following properties
• The digit in any position in the list has the
  same chance of being any of of 0 through 9
• The digits in different positions are
  independent, in that the value of one has no
  influence on the value of any other
• Any pair of digits has the same chance of being
  any of the 100 possible pairs, i.e., 00,01,02,
  ..98, 99
• Any triple of digits has the same chance of being
  any of the 1000 possible triples, i.e., 000, 001,
  002, 998, 999

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Using Random Number Tables

• A health inspector must select a SRS of size 5
  from 100 containers of ice cream to check for E.
  coli contamination
• The task is to draw a set of units from the
  sampling frame
• Assign a number to each individual
• Label the containers 00, 01,02,99
• Enter table and read across any line
• 81486 69487 60513 09297
• 81, 48, 66, 94, 87, 60, 51, 30, 92, 97

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Random Number Generation in R

• gt rnorm(10)
• gt rnorm(10, mean7, sd5)
• gt rbinom(10, size20, prob.5)
• We will revisit the meaning of the parameters at
  the end of todays session

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Combinatorics 1

• Back to draw five out of 40sample(140,5)
• The probability for any given number is 1/40 in
  the first sample,, 1/39 in the second, and so on
• ? P(x ) 1/(403938373635)
• gt 1/prod(4036)1 1.266449e-08
• But

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Combinatorics 2

• We dont care about the order of the five numbers
  out of 40
• There are 54321 combinations for the five
  drawn numbers
• ? gt prod(15) / prod(4036)1 1.519738e-06
• Shorthand for the above in gt 1/choose(40,5)

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Binomial Distribution

• Discrete probability distribution
• Events have only 2 possible outcomes
• binary, yes-no, presence-absence
• Computing probability of multiple events or
  trials
• Examples
• Probability that x number of people are alive at
  the age of 65
• Probability of a river reaching flood stage for
  three consecutive years

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When to Apply Binomial

• If sample is less than 10 of a large population
  in which a proportion p have acharacteristic of
  interest, then the distribution X, the number in
  the sample with that characteristic, is
  approximately binomial (n, p), where n is the
  sample size

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Geometric Distribution

• Tossing a biased coin until the first head
  appears pr(H) p
• pr(X x) pr(TTT H) pr(T1 n T2 n ..n Hx)
  (1 p)x-1 p

The geometric distribution is the distributionof
the number of tosses of a biased coin up toand
including the first head
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Poisson Distribution

• Discrete probability distribution
• Named in honor of Simeon Poisson (1781-1840)
• What is it used for?
• To model the frequency with which a specified
  event occurs over a period of time
• The specified event occurs randomly
• Independent of past or future occurrences
• Geographers also use this distribution to model
  how frequently an event occurs across a
  particular area
• We can also examine a data set (of frequency
  counts in order to determine whether a random
  distribution exists

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Poisson Distribution is used

• To analyze the number of patients arriving at a
  hospital emergency room between 6 AM and 7 AM on
  a particular day
• Obvious implications for resource allocation
• To analyze the number of phone calls per day
  arriving at a telephone switchboard
• To analyze the number of cars using the drive
  through window at a fast-food restaurant
• To analyze hailstorm occurrence in one Canadian
  province

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The Poisson Probability Formula
Lambda (?) is a positive real number (mean
frequency) e 2.718 (mathematical constant) X
0, 1, 2, 3, .(frequency of an occurrence) X! X
factorial
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Example - 1

• General Hospital, located in Phoenix, keeps
  records of emergency room traffic. From these
  records, we find that the number of patients
  arriving between 10 AM and 12 Noon has a Poisson
  distribution of with parameter
• ? 6.9
• Determine the probability that, on any given
  day, the number of patients arriving at that
  emergency room between 10 AM and 12 Noon will be
• Exactly four
• At the most two

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Exactly four arrivals, x4
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At the most, two arrivals
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Revisiting Mean and Standard Deviation
Mean
Standard Dev.
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What if

• We wanted to obtain a table of probabilities for
  the random variable X, the number of patients
  arriving between 10AM and 12 Noon?

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Discrete versus Continuous Distributions

• Moving from individual probabilities to total
  number of successes or failures
• Probability distribution f (x ) P (Xx) for
  discrete events
• Probability distribution for continuous events

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Expected Values

• Population standard deviationsquare root of the
  average squared distance of X from the mean m

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Expected Values

• Mean and Poisson distribution
• It can be shown that this adds to l. Thus, for
  Poisson-distributed populations E(X) l
• The standard deviation sd(X) for Poisson(l) is vl

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Probability Density Functions

• Moving from the discrete to the continuous
• Increasing the frequency of observations results
  in an ever finer histogram
• Total area under the curve 1

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Probability Density Functions

• Population means and standard devs
• mx balances the distribution
• The standard deviation is calculated as for
  discrete density functions

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The Normal Distribution
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Properties of a Normal Distribution

• Continuous Probability Distribution
• Symmetrical about a central point
• No skewness
• Central point in this dataset corresponds to all
  three measures of central tendency
• Also called a Bell Curve
• If we accept or assume that our data is normally
  distributed, then,
• We can compute the probability of different
  outcomes

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Properties of a Normal Distribution

• Using the symmetrical property of the
  distribution, we can conclude
• 50 of values must lie to the right, i.e. they
  are greater than the mean
• 50 of values must lie to the left, i.e.
• If the data is normally distributed, the
  probability values are also normally distributed
• The total area under the normal curve represents
  all (100) of probable outcomes
• What can you say about data values in a normally
  distributed data set?

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Normal Distribution and Standard Deviations
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Approximating a Normal Distribution

• In reality,
• If a variables distribution is shaped roughly
  like a normal curve,
• Then the variable approximates a normal
  distribution
• Normal Distribution is determined by
• Mean
• Standard Deviation
• These measures are considered parameters of a
  Normal Distribution / Normal Curve

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Equation of a Normal Curve
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Areas Within the Normal Curve

• For a normally distributed variable,
• the percentage of all possible observations that
  lie within any specified range equals the
  corresponding area under its associated normal
  curve expressed as a percentage
• A college has an enrollment of 3264 female
  students. Mean height is 64.4 inches, standard
  deviation is 2.4 inches
• Frequency and relative frequency are presented

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Frequency and Relative Frequency Table
0.0735, i.e. 7.35 of the students are between
67 and 68 inches tall
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Relative Frequency Histogram with Normal Curve
0.0735 the area that has been cross-hatched
Shaded area under the normal curve approximates
the percentage of students who are between 67-68
inches tall
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Standardizing a Normal Variable

• Once we have mean and standard deviation of a
  curve, we know its distribution and the
  associated normal curve
• Percentages for a normally distributed variable
  are equal to the areas under the associated
  normal curve
• There could be hundreds of different normal
  curves (one for each choice of mean or std. dev.
  value
• How can we find the areas under a standard normal
  curve?
• A normally distributed variable with a mean of 0
  and a standard deviation of 1 is said to have a
  standard normal distribution

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Z Score
The variable z is called the standardized version
of x, or the standardized variable corresponding
to x, with the mean 0 and standard deviation 1
Almost all observations in a dataset will lie
within three standard deviations to either side
of the mean, i.e., almost all possible
observations will have z scores between 3 and
3
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Normal Curve Properties

• The total area under the standard normal curve is
  equal to 1
• The standard normal curve extends infinitely in
  both directions, approaching but never touching
  the horizontal axis
• Standard normal curve is symmetric about 0
• Most of the area under a standard normal curve
  lies between 3 and 3

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Using the Standard Normal Table

• The times taken for runners to complete a local
  10 km race is normally distributed with a mean of
  61 minutes and a standard deviation of 9 minutes.
  Let x be the finish time of a randomly selected
  runner. Find the probability that x gt 75 minutes
• Step 1
• Calculate the standard score
• z 75-61/9 z 1.56
• Step 2
• Determine the probability from the normal
  table
• For z of 1.56, p 0.4406
• Step 3
• Interpret the result
• p (xgt75) 0.5 0.446
• 0.054 or 5.4 chance

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Using the Standard Normal Table

• In the previous example, what is the probability
  that someone finishes in less than 45 minutes?
• Step 1
• Calculate the standard score
• z 45-61/9 z -1.78
• Step 2
• Determine the probability from the normal
  table
• For z of -1.78, area 0.4625
• Step 3
• Interpret the result
• p (xlt45) 1- (0.50.4625)
• 0.038 or 3.8 of the runners finish in
  less than 45 minutes

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Three Distributions
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Normal Approximations for Discrete Distributions

• Approximation of the Binomial
• Binomial is used for large n and small p
• If p is moderate (not close to 0 or 1), then the
  Binomial can be approximated by the normal
• Rule of thumb np (1-p) 10
• Other normal approximations
• If X Poisson(l), normal works well for l 10

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Built-in Distributions in

• Four fundamental items can be calculated for a
  statistical distribution
• Density or point probability
• Cumulated probability, distribution function
• Quantiles
• Pseudo-random numbers
• In there are functions for each of these

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Density of a Normal Distribution

• gt x seq(-4, 4, 0.1)
• gt plot (x, dnorm(x), type"l")

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For Discrete Distributions..

• gt x 050
• gt plot (x, dbinom(x, size50, prob.33, type"h")

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Cumulative Distribution Functions

• Could be graphed but is not very informative
• Example
• Blood sugar concentration in the US population
  has a mean of 132 and a standard deviation of
  13.How special is a patient with a value 160?
• 1 pnorm(160, mean132, sd13)1 0.01562612
  or 1.5

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Random Number Generation in R

• gt rnorm(10)
• gt rnorm(10, mean7, sd5)
• gt rbinom(10, size20, prob.5)
• Now you understand the parameters