Probability Distributions

1
Probability Distributions

• GTECH 201
• Lecture 14

2
Probability Rules

• P lt 0 lt 1
• P (E) Frequency of that one outcome
• Number of possible outcomes
• for mutually exclusive, complementary events
• P (A) P (A) 1
• P (A) 1 P (A)
• P (A or B or C or.) P (A) P (B) P (C) ..
• P (A and B and C and ) P (A) P (B) P (C)
  
• where A, B, C are independent

3
When Events are NOT Independent

• P (A) or P (B) P (A) P (B) P (A B)
• We are selecting cards from a set of 52 playing
  cards. What is the probability that the card
  selected is either a spade or a face card
• P (spade)
• P (face card)
• P (spade and face card)
• Therefore P (spade) or p (face card)

4
Another Example

• 1990 arrest data shows that
• 79.6 of people arrested were male
• 18.3 were under 18 years
• 13.5 were males under 18 years
• If you select an inmate at random, what is the
  probability that the person is either male OR
  under 18?
• P (male or under 18) P (male) P (under 18)
  P (under 18)
• P (male) 0.796
• P (less than 18) 0.183
• P (male and less than 18) 0.135
• ? P (male or less than 18) 0.844

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Probability Distributions

• Over time, with enough data
• We recognize patterns
• Probability of outcomes are consistent
• E.g., tossing a coin, H or T
• Patterns are probability distributions
• Familiar with the bell curve
• For discrete outcomes
• Discrete probability distribution
• Continuous outcomes
• Continuous probability distribution

6
Binomial Distribution

• Discrete probability distribution
• Events have only 2 possible outcomes
• binary, yes-no, presence-absence
• Computing probability of multiple event
• P (x ) n! p x q n -x
• x !(n x )!
• where
• n number of events or trials
• p probability of the given (successful) outcome
  in a single trial
• q p bar (1-p)
• x number of times the given outcome occurs
  within n trials
• n! n factorial

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Factorial

• n factorial is written as n!
• Definition The factorial of a natural number n
  is the product of all non-zero numbers less than
  or equal n
• When n 0, 0! 1
• Therefore, when n 1,
• n! 1x(0)! or n! 1x(1) or
  1! 1
• When n 2, 2! 2x(1!)
• 2x(1), 2! 2
• When n 3, 3! 3x(2!)
• 3x(2)x(1), 3! 6
• For n gt 0, n! n (n -1)!
• n x (n -1) x (n -2)..x (2)
  x (1)
• Following this, when n 5, 5! ?

8
The Poisson Distribution

• Used to analyze how frequently an outcome occurs
  during a certain specified time period, or
  across a particular area
• Understanding the probability of events that
  occur randomly over time or space
• Revisit this distribution during the sessions on
  spatial statistics

9
Binomial Probability

• Assumptions
• n identical trials are to be performed
• Two outcomes, success or failure are possible
  for each trial
• The trials are independent
• The success probability p, remains the same
  from trial to trial

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To Find Binomial Probability

• Step 1 Identify a success
• Step 2 Determine p, the success probability
• Step 3 Determine n, the number of trials
• Step 4 Apply the binomial probability formula

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Example

• The National Institute of Mental Health reports
  that there is a 20 chance of an adult American
  suffering from a psychiatric disorder.
• Four randomly selected adult Americans are
  examined for psychiatric disorders.
• Find the probability that exactly three of the
  four people examined have a psychiatric disorder.

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Following the Steps Outlined

• Step 1 Identify a success
• i.e., selected individual suffers a psychiatric
  disorder
• Step 2 Probability of success
• p 0.2 Therefore q 1-0.2 0.8
• (Here failure selected individual does not
  suffer from a psychiatric disorder)
• Number of trials, n 4
• Now apply the formula

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Binomial Probability

• P (X3)
• 0.0256
• 2.56 chance that exactly 3 people
  selected at random will suffer from a
  psychiatric disorder

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See it Graphically
15
Calculate The Probability

• ssss (0.2)(0.2)(0.2)(0.2)
• sssf (0.2)(0.2)(0.2)(0.8)
• ssfs
• ssff

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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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Why do We Need Sampling ?

• Obvious reasons
• cost, practicality
• Accuracy
• Loss of the sample
• Issues related to undercounting
• Convenience sampling

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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

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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