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## Normal and Poisson Distributions

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Title: 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
• Sampling Frame
• The list of units from which the sample is chosen

3
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

5
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)

6
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

7
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

8
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

9
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

10
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)

11
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

12
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

13
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
14
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

15
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

16
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
17
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

18
Exactly four arrivals, x4
19
At the most, two arrivals
20
Revisiting Mean and Standard Deviation
Mean
Standard Dev.
21
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?

22
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

23
Expected Values
• Population standard deviationsquare root of the
average squared distance of X from the mean m

24
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

25
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

26
Probability Density Functions
• Population means and standard devs
• mx balances the distribution
• The standard deviation is calculated as for
discrete density functions

27
The Normal Distribution
28
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

29

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?

30
Normal Distribution and Standard Deviations
31
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

32
Equation of a Normal Curve
33
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

34
Frequency and Relative Frequency Table
0.0735, i.e. 7.35 of the students are between
67 and 68 inches tall
35
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
36
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

37
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
38
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

39
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

40
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

41
Three Distributions
42
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

43
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

44
Density of a Normal Distribution
• gt x seq(-4, 4, 0.1)
• gt plot (x, dnorm(x), type"l")

45
For Discrete Distributions..
• gt x 050
• gt plot (x, dbinom(x, size50, prob.33, type"h")

46
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

47
Random Number Generation in R
• gt rnorm(10)
• gt rnorm(10, mean7, sd5)
• gt rbinom(10, size20, prob.5)
• Now you understand the parameters