Transcript Probability Probability Distributions

Probability Distributions
GTECH 201
Lecture 14
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Probability Rules
P <= 0 <= 1
P (E) =
Frequency of that one outcome
Number of possible outcomes
for mutually exclusive, complementary events
P (A) + P (Ā) = 1
P (Ā) = 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
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When Events are NOT Independent
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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) =
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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
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Over time, with enough data
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Patterns are probability distributions
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Familiar with the bell curve
For discrete outcomes
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We recognize patterns
Probability of outcomes are consistent
E.g., tossing a coin, H or T
Discrete probability distribution
Continuous outcomes
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Continuous probability distribution
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Binomial Distribution
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Discrete probability distribution
Events have only 2 possible outcomes
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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
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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
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Therefore, when n = 1,
n! = 1x(0)! or
When n = 2,
2! = 2x(1!)
= [2x(1)],
When n = 3,
3! = 3x(2!)
= 3x(2)x(1),
For n > 0,
n! = n (n -1)!
= [n x (n -1)
Following this, when n = 5,
5! = ?
n! = 1x(1) or 1! = 1
2! = 2
3! = 6
x (n -2)………..x (2) x (1)]
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The Poisson Distribution
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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
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Binomial Probability
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Assumptions
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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
Step
Step
Step
1
2
3
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Identify a success
Determine p, the success probability
Determine n, the number of trials
Apply the binomial probability formula
X
( n x )
n! p q
P X  
X !n  X !
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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
X
( n x )
n! p q
P X  
X !n  X !
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Binomial Probability
=
4!0.2 0.8
3!4  3!
=
4  3  2 10.0080.8
3  2 11
=
0.0256
=
2.56 % chance that exactly 3 people
selected at random will suffer
from a psychiatric disorder
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P (X=3)
4 3
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See it Graphically
0.2
0.2
S
0.2
0.2
0.8
F
0.8
0.8
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Calculate The Probability
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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
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Population
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Unit
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any individual member of the population
Sample
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The entire group of objects about which
information is sought
a part or a subset of the population used to gain
information about the whole
Sampling Frame
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The list of units from which the sample is chosen
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Why do We Need Sampling ?
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Obvious reasons
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cost, practicality
Accuracy
Loss of the sample
Issues related to undercounting
Convenience sampling
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Simple Random Sampling
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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
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A table of random digits is:
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A list of 10 digits 0 through 9 having the following
properties
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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
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A health inspector must select a SRS of size 5
from 100 containers of ice cream to check for
E. coli contamination
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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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