Probability Terminology Classical Interpretation Notion of probability based on equal likelihood of individual possibilities coin toss has 12 chance of Heads card draw has 452 chance of an Ace Origins in games of chance ID: 657806
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Slide1
Chapter 4
Basic Probability and Probability DistributionsSlide2
Probability Terminology
Classical Interpretation
: Notion of probability based on equal likelihood of individual possibilities (coin toss has 1/2 chance of Heads, card draw has 4/52 chance of an Ace). Origins in games of chance.
Outcome
: Distinct result
of random process (
N
= # outcomes)
Event
: Collection of outcomes (
N
e
= # of outcomes in event)
Probability of event
E
:
P
(event
E
) =
N
e
/
N
Relative Frequency Interpretation
: If an experiment were conducted repeatedly, what fraction of time would event of interest occur (based on empirical observation)
Subjective Interpretation
: Personal view (possibly based on external info) of how likely a one-shot experiment will end in event of interest Slide3
Obtaining Event Probabilities
Classical Approach
List all
N
possible outcomes of experiment
List all
N
e
outcomes corresponding to event of interest (
E
)
P
(event
E
) =
N
e
/
N
Relative Frequency Approach
Define event of interest
Conduct experiment repeatedly (often using computer)
Measure the fraction of time event
E
occurs
Subjective Approach
Obtain as much information on process as possible
Consider different outcomes and their likelihood
When possible, monitor your skill (e.g. stocks, weather)Slide4
Basic Counting RulesSlide5
Basic Probability and Rules
A,B
Events of interest
P
(
A
),
P
(
B
) Event probabilities
Union
: Event
either
A
or
B
occurs
(
A
B
)
Mutually Exclusive
:
A, B
cannot occur at same time
If
A,B
are mutually exclusive:
P
(either
A
or
B
) =
P
(
A
) +
P
(
B
)
Complement of
A
: Event that
A
does not occur (
Ā
)
P
(
Ā
) = 1-
P
(
A
) That is:
P
(
A
) +
P
(
Ā
) = 1
Intersection
: Event
both
A
and
B
occur
(
A
B
or
AB
)
P
(
A
B
) =
P
(
A
) +
P
(
B
) -
P
(
AB
) Slide6
Conditional Probability and Independence
Unconditional/Marginal Probability
: Frequency which event occurs in general (given no additional info).
P
(
A
)
Conditional Probability
: Probability an event (
A
) occurs
given
knowledge another event (B) has occurred. P(A|B)Independent Events: Events whose unconditional and conditional (given the other) probabilities are the sameSlide7
John Snow London Cholera Death Study
2 Water Companies (Let
D
be the event of death):
Southwark&Vauxhall (
S
): 264913 customers, 3702 deaths
Lambeth (
L
): 171363 customers, 407 deaths
Overall: 436276 customers, 4109 deaths
Note that probability of death is almost 6 times higher for S&V customers than Lambeth customers (was important in showing how cholera spread)Slide8
John Snow London Cholera Death Study
Contingency Table with joint probabilities (in body of table) and marginal probabilities (on edge of table)Slide9
John Snow London Cholera Death Study
WaterUser
S&V
L
.6072
.3928
Company
Death
D (.0085)
.0140
.9860
D
C
(.5987)
.0024
.9976
D (.0009)
D
C
(.3919)
Tree Diagram obtaining joint probabilities by multiplication ruleSlide10
Bayes’s Rule - Updating Probabilities
Let
A
1
,…,
A
k
be a set of events that
partition
a sample space such that (mutually exclusive and exhaustive):
each set has known
P(A
i) > 0 (each event can occur)for any 2 sets Ai and Aj, P(Ai
and Aj) = 0 (events are disjoint)P(A1) + … +
P(A
k
) =
1 (each outcome belongs to one of events)
If
C
is an event such that
0 <
P(C)
< 1 (
C
can occur, but will not necessarily occur)
We know the probability will occur given each event
Ai
:
P(C|A
i
)Then we can compute probability of Ai given C occurred:Slide11
Northern Army at Gettysburg
Regiments: partition of soldiers (
A
1
,…,
A
9
). Casualty: event
C
P(A
i
) = (size of regiment) / (total soldiers) = (Column 3)/95369
P(C|A
i
) =
(# casualties) / (regiment size) = (Col 4)/(Col 3)
P(C|A
i
) P(A
i
) =
P(A
i
and
C)
= (Col 5)*(Col 6)
P(C)
=sum(Col 7)
P(A
i|C) =
P(Ai and C) / P(C) = (Col 7)/.2416Slide12
Example - OJ Simpson Trial
Given Information on Blood Test (
T
+/
T
-)
Sensitivity:
P
(
T
+|Guilty)=1
Specificity:
P(T-|Innocent)=.9957 P(T+|I)=.0043
Suppose you have a prior belief of guilt: P(G)=p*What is “posterior” probability of guilt after seeing evidence that blood matches:
P
(
G
|
T
+)?
Source: B.Forst (1996). “Evidence, Probabilities and Legal Standards for Determination of Guilt: Beyond the OJ Trial”, in
Representing OJ: Murder, Criminal Justice, and the Mass Culture
, ed. G. Barak pp. 22-28. Harrow and Heston, Guilderland, NYSlide13Slide14
Random Variables/Probability Distributions
Random Variable
: Outcome characteristic that is not known prior to experiment/observation
Qualitative Variables
: Characteristics that are non-numeric (e.g. gender, race, religion, severity)
Quantitative Variables
: Characteristics that are numeric (e.g. height, weight, distance)
Discrete
: Takes on only a countable set of possible values
Continuous
: Takes on values along a continuum
Probability Distribution
: Numeric description of outcomes of a random variable takes on, and their corresponding probabilities (discrete) or densities (continuous)Slide15
Discrete Random Variables
Discrete RV: Can take on a finite (or countably infinite) set of possible outcomes
Probability Distribution: List of values a random variable can take on and their corresponding probabilities
Individual probabilities must lie between 0 and 1
Probabilities sum to 1
Notation:
Random variable:
Y
Values
Y
can take on:
y
1, y2, …, ykProbabilities: P(
Y=y1) = p1 …
P
(
Y
=
y
k
) =
p
k
p
1
+ … +
pk = 1Slide16
Example: Wars Begun by Year (1482-1939)
Distribution of Numbers of wars started by year
Y
= # of wars stared in randomly selected year
Levels:
y
1
=0,
y
2
=1,
y
3=2, y4=3, y5=4Probability Distribution:Slide17
Masters Golf Tournament 1st Round ScoresSlide18
Means and Variances of Random Variables
Mean: Long-run average
a random variable will take on (also the balance point of the probability distribution)
Expected Value is another term, however we really do not expect that a realization of
X
will necessarily be close to its mean. Notation:
E
(
X
)
Mean and Variance of a discrete random variable:Slide19
Rules for Means
Linear Transformations:
a
+
bY
(where
a
and
b
are constants):
E
(
a+bY) = ma+bY = a + bmYSums of random variables: X + Y (where X and
Y are random variables): E(X+Y) = m
X+Y
=
m
X
+
m
Y
Linear Functions of Random Variables:
E
(
a
1
Y
1
+
+
anYn) =
a1m1
+…+anmn where E(Yi)=miSlide20
Example: Masters Golf Tournament
Mean by Round (Note ordering):
m
1
=73.54
m
2
=
73.07
m
3
=73.76
m4=73.91Mean Score per hole (18) for round 1: E((1/18)X
1) = (1/18)m1 = (1/18)73.54 = 4.09Mean Score versus par (72) for round 1:
E
(
X
1
-72) =
m
X1-72
= 73.54-72= +1.54 (1.54 over par)
Mean Difference (Round 1 - Round 4):
E
(
X
1
-
X
4) = m1 - m4 = 73.54 - 73.91 = -0.37
Mean Total Score: E(X1+X
2+X3+X4) = m1+ m2+ m3+ m4 = = 73.54+73.07+73.76+73.91 = 294.28 (6.28 over par)Slide21
Variance of a Random Variable
Special Cases:
X
and
Y
are independent (outcome of one does not alter the distribution of the other):
r
= 0, last term drops out
a=b=
1 and
r
= 0
V
(
X+Y
) =
s
X
2
+
s
Y
2
a=1 b= -
1 and
r
= 0
V
(
X-Y
) =
s
X2 +
sY2 a=b=1 and r 0 V(X+Y) = sX2 + sY2 + 2
rsXsY a=1 b= -1 and r 0 V(X-Y) = sX2 + sY
2 -2rsX
sYSlide22
Examples - Wars & Masters Golf
m
=0.67
m
=73.54Slide23
Binomial Distribution for Sample Counts
Binomial “Experiment”
Consists of
n
trials or observations
Trials/observations are independent of one another
Each trial/observation can end in one of two possible outcomes often labelled “Success” and “Failure”
The probability of success,
p
, is constant across trials/observations
Random variable,
Y
, is the number of successes observed in the n trials/observations. Binomial Distributions: Family of distributions for Y, indexed by Success probability (p) and number of trials/observations (
n). Notation: Y~B(n,p)Slide24
Binomial Distributions and Sampling
Problem when sampling from a finite population: the sequence of probabilities of Success is altered after observing earlier individuals.
When the population is much larger than the sample (say at least 20 times as large), the effect is minimal and we say
X
is approximately binomial
Obtaining probabilities:Slide25
Example - Diagnostic Test
Test claims to have a sensitivity of 90% (Among people with condition, probability of testing positive is .90)
10 people who are known to have condition are identified,
Y
is the number that correctly test positive
Table obtained in EXCEL with function:
BINOM.DIST(
k,n,
p
,
FALSE
)
(TRUE option gives cumulative distribution function:
P
(
Y
k
)Slide26
Binomial Mean & Standard Deviation
Let
S
i
=1 if the i
th
individual was a success, 0 otherwise
Then
P
(
S
i
=1) = p and P(Si=0) = 1-pThen E
(Si)=mS = 1(
p
) + 0(1-
p
) =
p
Note that
Y = S
1
+…+
S
n
and that trials are independent
Then
E
(
Y
)=
mY = nmS = n
pV(Si) = E
(Si2)-mS2 = p-p2 = p(1-p)Then V(Y)=sY2 = np
(1-p)Slide27
Poisson Distribution for Event Counts
Distribution related to Binomial for Counts of number of events occurring in fixed time or space. Takes many “sub-intervals” and assumes Binomial (
n
=1) distribution for events in each.
The average number of events in unit time or space is
m
. In general, for length
t
, mean is
m
t
Table 14 (pp. 1121-1123) gives probabilities for selected
m
EXCEL Function: =POISSON.DIST(
y
,
m
,0) returns
P
(
y
)Slide28
Poisson Distribution for Event CountsSlide29
Continuous Random Variables
Variable can take on any value along a continuous range of numbers (interval)
Probability distribution is described by a smooth density curve
Probabilities of ranges of values for
Y
correspond to areas under the density curve
Curve must lie on or above the horizontal axis
Total area under the curve is 1
Special cases: Normal and Gamma distributionsSlide30
Normal Distribution
Bell-shaped, symmetric family of distributions
Classified by 2 parameters: Mean (
m
) and standard deviation (
s
). These represent
location
and
spread
Random variables that are approximately normal have the following properties wrt individual measurements:
Approximately half (50%) fall above (and below) mean
Approximately 68% fall within 1 standard deviation of meanApproximately 95% fall within 2 standard deviations of meanVirtually all fall within 3 standard deviations of meanNotation when Y is normally distributed with mean m
and standard deviation s :Slide31
Two Normal DistributionsSlide32
Normal DistributionSlide33
Example - Heights of U.S. Adults
Female and Male adult heights are well approximated by normal distributions:
Y
F
~N(63.7,2.5)
Y
M
~N(69.1,2.6)
Source:
Statistical Abstract of the U.S.
(1992)Slide34
Standard Normal (
Z
) Distribution
Problem: Unlimited number of possible normal distributions (-
<
m
< ,
s
> 0)
Solution: Standardize the random variable to have mean 0 and standard deviation 1
Probabilities of certain ranges of values and specific percentiles of interest can be obtained through the standard normal (
Z
) distributionSlide35Slide36
Standard Normal (Z) Distribution
z
Table Area
1-Table AreaSlide37
I
n
t
g
e
r
p
a
r
t
&
1st
D
e
c
i
m
a
l
2nd Decimal PlaceSlide38
2nd Decimal Place
I
n
t
g
e
r
p
a
r
t
&
1st
D
e
c
i
m
a
lSlide39
Finding Probabilities of Specific Ranges
Step 1 -
Identify the normal distribution of interest (e.g. its mean (
m
) and standard deviation (
s
) )
Step 2 -
Identify the range of values that you wish to determine the probability of observing (
y
L
,
yU), where often the upper or lower bounds are or -Step 3 - Transform y
L and yU into
Z
-values:
Step 4 -
Obtain P(
z
L
Z
z
U
) from
Z
-table (pp. 1086-7)Slide40
Example - Adult Female Heights
What is the probability a randomly selected female is 5’10” or taller (70 inches)?
Step 1 -
Y
~ N(63.7 , 2.5)
Step 2 -
y
L
= 70.0
y
U = Step 3 -
Step 4 -
P(
Y
70) = P(
Z
2.52) =
1-P(Z2.52)=1-.9941=.0059 ( 1/170) Slide41
Finding Percentiles of a Distribution
Step 1 -
Identify the normal distribution of interest
(e.g. its mean (
m
) and standard deviation (
s
) )
Step 2
-
Determine the percentile of interest 100
p% (e.g. the 90th percentile is the cut-off where only 90% of scores are below and 10% are above).Step 3 - Find p in the body of the z-table and itscorresponding z-value
(zp) on the outer edge:If 100p < 50 then use left-hand page of table
If 100
p
50 then use right-hand page of table
Step 4
- Transform
z
p
back to original units:Slide42
Example - Adult Male Heights
Above what height do the tallest 5% of males lie above?
Step 1 -
Y
~ N(69.1 , 2.6)
Step 2 -
Want to determine 95
th
percentile (
p
= .95)
Step 3 -
P(Z1.645) = .95Step 4 - y.95 = 69.1 + (1.645)(2.6) = 73.4 (6’,1.4”)Slide43
Assessing Normality and Transformations
Obtain a histogram and see if mound-shaped
Obtain a normal probability plot
Order data from smallest to largest and rank them (1 to
n
)
Obtain a percentile for each: pct = (rank-0.375)/(
n
+0.25)
Obtain the
z
-score corresponding to the percentile
Plot observed data versus z-score, see if straight line (approx.)Transformations that can achieve approximate normality:Slide44
Chi-Square Distribution
Indexed by “degrees of freedom (
n
)” X~
c
n
2
Z~N(0,1)
Z
2
~
c
12Assuming Independence:
Obtaining Probabilities in EXCEL:
To obtain: 1-F(x)=P(
X≥x
) Use Function: =CHISQ.DIST.RT(
x,
n
)
Table 7 (pp. 1095-6) Gives critical values for selected upper tail probabilitiesSlide45
Chi-Square DistributionsSlide46
Critical Values for Chi-Square Distributions (Mean=
n
, Variance=2
n
)Slide47
Student’s t-Distribution
Indexed by “degrees of freedom (
n
)” X~t
n
Z~N(0,1), X~
c
n
2
Assuming Independence of Z and X:
Obtaining Probabilities in EXCEL:
To obtain: 1-F(t)=P(
T≥t
) Use Function: =T.DIST.RT(
t,
n
)
Table 2 (p. 1088 gives critical values for selected upper tail
probsSlide48Slide49
Critical Values for Student’s t-DistributionsSlide50
F-Distribution
Indexed by 2 “degrees of freedom (
n
1
,n
2
)” W~F
n1,n2
X
1
~
c
n12, X2
~cn22Assuming Independence of X
1
and X
2
:
Obtaining Probabilities in EXCEL:
To obtain: 1-F(w)=P(
W≥w
) Use Function: =F.DIST.RT(w,
n
1
,n
2
)
Table 8 (pp. 1097-1108) gives critical values for selected upper tail
probsSlide51Slide52
Critical Values for F-distributions P(F ≤ Table Value) = 0.95Slide53
Sampling Distributions
Distribution of a Sample Statistic: The probability distribution of a sample statistic obtained from a random sample or a randomized experiment
What values can a sample mean (or proportion) take on and how likely are ranges of values?
Population Distribution: Set of values for a variable for a population of individuals. Conceptually equivalent to probability distribution in sense of selecting an individual at random and observing their value of the variable of interestSlide54
Sampling Distribution of a Sample Mean
Obtain a sample of
n
independent measurements of a quantitative variable:
Y
1
,…,
Y
n
from a population with mean
m
and standard deviation
sAverages will be less variable than the individual measurementsSampling distributions of averages will become more like a normal distribution as n increases (regardless of the shape of the population of individual measurements)Slide55
Central Limit Theorem
When random samples of size
n
are selected from any population with mean
m
and finite standard deviation
s
, the sampling distribution of the sample mean will be approximately distributed for large
n
:
Z-
table can be used to approximate probabilities of ranges of values for sample means, as well as percentiles of their sampling distribution Slide56
Sample Proportions
Counts of Successes (
Y
) rarely reported due to dependency on sample size (
n
)
More common is to report the
sample proportion
of successes:Slide57
Sampling Distributions for Counts & Proportions
For samples of size
n
, counts (and thus proportions) can take on only
n
distinct possible outcomes
As the sample size
n
gets large, so do the number of possible values, and sampling distribution begins to approximate a normal distribution. Common Rule of thumb:
n
p
10 and n(1-p) 10 to use normal approximationSlide58
Sampling Distribution for
Y
~
B
(
n
=1000,
p=0.2)Slide59
Using
Z
-Table for Approximate Probabilities
To find probabilities of certain ranges of counts or proportions, can make use of fact that the sample counts and proportions are approximately normally distributed for large sample sizes.
Define range of interest
Obtain mean of the sampling distribution
Obtain standard deviation of sampling distribution
Transform range of interest to range of
Z
-values
Obtain (approximate) Probabilities from
Z-
table