Normal and Poisson Distributions

- GTECH 201
- Lecture 14

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

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

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

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)

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

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

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

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

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)

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

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

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

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

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

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

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

Exactly four arrivals, x4

At the most, two arrivals

Revisiting Mean and Standard Deviation

Mean

Standard Dev.

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?

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

Expected Values

- Population standard deviationsquare root of the

average squared distance of X from the mean m

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

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

Probability Density Functions

- Population means and standard devs
- mx balances the distribution
- The standard deviation is calculated as for

discrete density functions

The Normal Distribution

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

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?

Normal Distribution and Standard Deviations

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

Equation of a Normal Curve

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

Frequency and Relative Frequency Table

0.0735, i.e. 7.35 of the students are between

67 and 68 inches tall

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

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

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

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

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

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

Three Distributions

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

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

Density of a Normal Distribution

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

For Discrete Distributions..

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

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

Random Number Generation in R

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