1. Normal and Poisson Distributions
GTECH 201
Lecture 14

2. Sampling Population Unit Sample Sampling Frame
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

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 1:40, you can > sample(1:40,5) [1]

5. Sampling with Replacement
Default in R is ‘without replacement’ sample(c("H", "T"), 10, replace=T) [1] "T" "T" "T" "T" "T" "H" "H" "T" "H" "T“ prob=c(.9,.1)
sample(c("S", "F"), 10, replace=T, 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
81, 48, 66, 94, 87, 60, 51, 30, 92, 97

8. Random Number Generation in R
> rnorm(10)
> rnorm(10, mean=7, sd=5)
> rbinom(10, size=20, prob=.5)
We will revisit the meaning of the parameters at the end of today’s session

9. Combinatorics 1 Back to draw five out of 40 sample(1:40,5)
The probability for any given number is 1/40 in the first sample,, 1/39 in the second, and so on
 P(x ) = 1/(40*39*38*37*36*35)
> 1/prod(40:36) [1] e-08
But…

10. Combinatorics 2
We don’t care about the order of the five numbers out of 40
There are 5*4*3*2*1 combinations for the five drawn numbers
 > prod(1:5) / prod(40:36) [1] e-06
Shorthand for the above in > 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 a characteristic 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(TT…T H) = pr(T1 ∩ T2 ∩ ..∩ Hx) = (1 – p)x-1 p
The geometric distribution is the distribution of the number of tosses of a biased coin up to and including the first head

14. Poisson Distribution Discrete probability distribution
Named in honor of Simeon Poisson ( )
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 = (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, x=4

19. At the most, two arrivals…

20. Revisiting Mean and Standard Deviation

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 (X=x) for discrete events:
Probability distribution for continuous events:

23. Expected Values
Population standard deviation square 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 √l

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 dev’s
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 variable’s 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
Mean = ; Standard Deviation =
e = ; = 3.142

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. Relative Frequency Table
Frequency and
Relative Frequency Table
0.0735, i.e % of the students are between 67 and 68 inches tall

35. Relative Frequency Histogram with Normal Curve
= the area that has been cross-hatched
Shaded area under the normal curve approximates the percentage of students who are between 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 > 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 =
Step 3
Interpret the result
p (x>75) = 0.5 – 0.446
= 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=
Step 3
Interpret the result
p (x<45) = 1- ( )
= 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
> x = seq(-4, 4, 0.1)
> plot (x, dnorm(x), type="l")

45. For Discrete Distributions..
> x = 0:50
> plot (x, dbinom(x, size=50, 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, mean=132, sd=13) [1] or 1.5%

47. Random Number Generation in R
> rnorm(10)
> rnorm(10, mean=7, sd=5)
> rbinom(10, size=20, prob=.5)
Now you understand the parameters…