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Mean = 41.21 Median = 42.5 s = 7.59
x - 1s–
x + 1s–
Assignment #1
Course Schedule
Probabilities in Geography
• The analyses of many problems (daily or
geographic) are often based on probabilities, such
as:
• What are the “chances” of having rain over the
weekend?
• What is the “likelihood” that the 100-year flood will
occur within the next ten years?
• How “likely” is it that a pixel on a satellite image is
correctly classified or misclassified?
Probability & Probability Distribution
• We summarize a sample statistically and want to make some inferences about the population (e.g., what proportion of the population has values within a given range)
• The concept of probability is the key to making statistical inferences by sampling a population
• What we are doing is trying to ascertain the probability of an event having a given outcome
• This requires us to be able to specify the distribution of a variable before we can make inferences
Probability & Probability Distributions
• Previously, we looked at some proportions of area under the normal curve:
Source: Earickson, RJ, and Harlin, JM. 1994. Geographic Measurement and Quantitative Analysis. USA: Macmillan College Publishing Co., p. 100.
Probability & Probability Distributions
• BUT before we could use the normal curve, we have to find out if this is the right distribution for our variable …
• While many natural phenomena are normally distributed, there are other phenomena that are best described using other distributions
• Background on probabilities (terminology & rules), and a few useful distributions:
• Discrete distributions: Binomial and Poisson
• Continuous distributions: Normal and its relatives
Probability-Related Concepts
• An event – Any phenomenon you can observe that can have more than one outcome (e.g., flipping a coin)
• An outcome – Any unique condition that can be the result of an event (e.g., flipping a coin: heads or tails), a.k.a simple event or sample points
• Sample space – The set of all possible outcomes associated with an event
– e.g., flip a coin – heads (H) and tails (T)
– e.g., flip a coin twice – HH, HT, TH, TT
• Associated with each possible outcome in a
sample space is a probability
• Probability is a measure of the likelihood of
each possible outcome
• Probability measures the degree of uncertainty
• Each of the probabilities is greater than or equal
to zero, and less than or equal to one
• The sum of probabilities over the sample space
is equal to one
Probability-Related Concepts
Probability – Examples
• Example I – Flip a coin
– Two possible outcomes: “heads”, “tails”
– Each outcome is equally likely
– “heads” and “tails” have the same probability
(0.5)
– The sum of probabilities over the sample
space is one
– # of “heads” and # of “tails” will be nearly equal
Probability – Examples
• Example II – Flip a coin twice– Four outcomes are equally likely
– Tosses of the coin are independent
– Each outcome has probability 1/4
– The probability of a head on Flip 1 and a head on Flip 2 is 1/2 * 1/2 = 1/4
Outcome First flip Second flip
1 Heads Heads
2 Heads Tails
3 Tails Heads
4 Tails Tails
How To Assign Probabilities to Experimental Outcomes?
• There are numerous ways to assign probabilities to the elements of sample spaces
• Classical method assigns probabilities based on the assumption of equally likely outcomes
• Relative frequency method assigns probabilities based on experimentation or historical data
• Subjective method assigns probabilities based on the assignor’s judgment or belief
Classical Method
• This approach assumes that each outcome is
equally likely
• If an experiment has n possible outcomes, this
method would assign a probability of 1/n to each
outcome.
• It is an appropriate way to assign probabilities to
the outcomes in special kinds of experiments
Classical Method
• Example I: Rolling a die
• Sample Space: S = {1, 2, 3, 4, 5, 6}
• Probabilities: Each sample point has a 1/6
chance of occurring.
Classical Method
• Example II – Flip four coins– Let “0” represent “heads” and “1” represents “tails”– For each toss, the probability of “heads” or “tails” is ½– Assuming that outcomes of the four tosses are
independent from one another– Sixteen possible outcomes
× × × ×
½ ½ ½ ½ Probability of each outcome:
½ * ½ * ½ * ½ = 1/16 = 0.0625
0000 0100 1000 1100
0001 0101 1001 1101
0010 0110 1010 1110
0011 0111 1011 1111
Relative Frequency Method
• The second way is to assign them on the basis of relative frequencies
• Example
– Given a weather pattern, a meteorologist may note that in 65 out of the last 100 times that such a pattern prevailed there was measurable precipitation the next day
– If there were such a weather pattern today, what would the probability of having rain tomorrow be?
– The possible outcomes – rain or no rain tomorrow – are assigned probabilities of 0.65 and 0.35, respectively
Subjective Method
• When extreme weather conditions occur it might be inappropriate to assign probabilities based solely on historical data
• We can use any data available as well as our experience and intuition, but ultimately a probability value should express our degree of belief that the experimental outcome will occur
• The best probability estimates often are obtained by combining the estimates from the classical or relative frequency approach with the subjective estimates.
Probability Rules
• Rules for combining multiple probabilities• A useful aid is the Venn diagram - depicts multiple
probabilities and their relations using a graphical depiction of sets
• The rectangle that forms the area of the Venn Diagram represents the sample (or probability) space, which we have defined above
• Figures that appear within the sample space are sets that represent events in the probability context, & their area is proportional to their probability (full sample space = 1)
A B
Probability Rules• We can use a Venn diagram to describe the
relationships between two sets or events, and the corresponding probabilities
•The union of sets A and B (written symbolically is A B) is represented by the areas enclosed by set A and B together, and can be expressed by OR (i.e. the union of the two sets includes any location in A or B)
•The intersection of sets A and B (written symbolically as A B) is the area that is overlapped by both the A and B sets, and can be expressed by AND (i.e. the intersection of the two sets includes locations in A AND B)
A B
A B
Addition Rule• If sets A and B do not overlap in the Venn diagram,
the sets are disjoint, and this represents a case of two independent, mutually exclusive events
•The union of sets A and B here uses the addition rule, where
P(A = P(A) + P(B)
•You can think of this in terms of areas of the events, where the union in this case is simply the sum of the areas
•The intersection of sets A and B here results in the empty set (symbolized by ), because at no point do the circles overlap
A B
A B
P(A = P(A) + P(B)
P(A =
Probability Rules
• The union of sets A and B here uses the addition rule, where
P(A = P(A) + P(B)
P(A = 2/6 + 2/6
P(A = 4/6 = 2/3 = 0.67
•The outcomes represented here are mutually exclusive, thus there is no intersection between sets A and B, thus P(A =
A B
A B
P(A = P(A) + P(B)
P(A =
• For example, suppose set A represents a roll of 1 or 2 on a 6-sided die, so P(A)=2/6, and set B represents a roll of 3 or 4, so P(B)=2/6
Probability Rules – General Addition Rule
• If sets A and B do overlap in the Venn diagram, the sets are independent but not mutually exclusive
•The union of sets A and B here isP(A = P(A) + P(B) - P(A
because we do not wish to count the intersection area twice, thus we need to subtract it from the sum of the areas of A and B when taking the union of a pair of overlapping sets
The intersection of sets A and B here is calculated by taking the product of the two probabilities, a.k.a. the multiplication rule:
A B
A B
P(A = P(A) * P(B)
P(A = P(A) + P(B) - P(A
General Addition Rule• Consider set A to give the chance of precipitation
at P(A)=0.4 and set B to give the chance of below freezing temperatures at P(B)=0.7
•The intersection of sets A and B here is P(A = P(A) * P(B)
P(A = 0.4 * 0.7 = 0.28
This expresses the chance of snow at P(A = 0.28
•The union of sets A and B here is
P(A = P(A) + P(B) - P(A
P(A = 0.4 + 0.7 – 0.28 = 0.82
This expresses the chance of below freezing temperatures or precipitation occurring at P(A = 0.82
A B
P(A = P(A) + P(B) - P(A
A B
P(A = P(A) * P(B)
Complement• Consider set A to give the chance of precipitation
at P(A)=0.4 and set B to give the chance of below freezing temperatures at P(B)=0.7
•The complement of set A is
P(A’ = 1 - P(A)
P(A’ = 1 – 0.4 = 0.6
This expresses the chance of it not raining or snowing at P(A’ = 0.6
•The complement of the union of sets A and B is
P(A’ = 1 – [P(A) + P(B) - P(AP(A’ = 1 – [0.4 + 0.7 – 0.28] = 0.18
This expresses chance of it neither raining nor being below freezing at P(A’ = 0.18
P(A’ = 1 - P(A)
P(A’ = 1 – [P(A) + P(B) - P(A
A A’
A BP
(A
’
Probability Rules• We can also encounter the situation where set A is
fully contained within set B, which is equivalent to saying that set A is a subset of set B:
• For example, set A might represent precipitation events with >= 5 inches, whereas set B denotes any events with >= 1 inch A is contained with B because anytime A occurs, B occurs as well
• In probability terms, this situation
occurs when outcome B is a
necessary precondition for outcome
A to occur, although not vice-versa
(in which case set B would be
contained in set A instead)
A
B
Probability – Example
• Example – # of malls within cities City # of Malls
A 1 B 4 C 4 D 4 E 2 F 3
• We might wonder if we randomly pick one of these six cities, what is the probability (chance) that it will have n malls?
SampleSpace
Each count of the # of malls in a city is an event
Random Variables
• What we have here is a random variable – defined as a function that associates a unique numerical value with every outcome of an experiment
• To put this another way, a random variable is a function defined on the sample space this means that we are interested in all the possible outcomes
• A random variable X is a rule that assigns a numerical value to each outcome in the sample space of an experiment
Random Variables
• The value of the random variable will vary from trial to trial as the experiment is repeated
• We use an uppercase letter to denote a random variable and a lowercase letter to denote a particular value of the variable
• A random variable can be classified as being either discrete or continuous depending on the numerical values it assumes
Discrete & Continuous Variables
• Discrete variable – A variable that can take on
only a finite number of values– # of malls within cities
– # of vegetation types within geographic regions
– # population
• Continuous variable – A variable that can take
on an infinite number of values (all real number
values)– Elevation (e.g., [500.0, 1000.0])
– Temperature (e.g., [10.0, 20.0])
– Precipitation (e.g., [100.0, 500.0]
Probability Distribution & Probability Function
• The question was: If we randomly pick one of the six cities, what is the probability (or chance) that it will have n malls?
• To answer this question, we need to form a probability function (probability distribution) from the sample space that gives all values of a random variable and their probabilities
• Then we can find the probability that a randomly selected city has n malls from the probability function
Probability Function & Probability Distribution
• The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable
• In other words, a probability distribution expresses the relative number of times we expect a random variable to assume each and every possible value
• The probability distribution of a random variable may be represented by a table, a graph, or an equation
Probability Function & Probability Distribution
• The probability distribution is defined by a probability function, denoted by p(X) or f(x), which provides the probability for each value of the random variable
• p(X) or f(x) represents the probability function or the probability distribution for the random variable X
Probability Function – An Example• Here, the values of xi are drawn from the four
outcomes, and their probabilities are the number of events with each outcome divided by the total number of events:
City # of Malls
A 1
B 4
C 4
D 4
E 2
F 3
xi P(xi)
1 1/6 = 0.1672 1/6 = 0.1673 1/6 = 0.1674 3/6 = 0.5
• The probability of an outcome P(xi) = # of times an outcome occurred
Total number of events
Probability Function
• We can plot this probability distribution as a probability function:
• This plot uses thin lines to denote that the probabilities are massed at discrete values of this random variable
xi p(xi)
1 1/6 = 0.1672 1/6 = 0.1673 1/6 = 0.1674 3/6 = 0.5 0
0.25
0.50
p(x i)
1 2 3 4
xi
Probability Mass Functions
• A discrete random variable can be described by a
probability mass function (pmf)
• A probability mass function is usually represented
by a table, graph, or equation
• The probability of any outcome must satisfy:
0 <= p(X=xi) <= 1 i = 1, 2, 3, …, k-1, k
• The sum of all probabilities in the sample space
must total one, i.e.1)(
1
k
iixXp
Probability Mass Function
• Example: # of malls in cities
• This plot uses thin lines to denote that the probabilities are massed at discrete values of this random variable
xi p(X=xi)
1 1/6 = 0.1672 1/6 = 0.1673 1/6 = 0.1674 3/6 = 0.5
0
0.25
0.50
p(x i)
1 2 3 4
xi
Discrete Probability Distribution
• We can calculate the mean and variance of a discrete probability distribution:
• We use µ and σ2 here because the basic idea of a probability distribution is to use a large number of samples to approach the distribution of a population
xi *p(xi)i=1
i=k
(xi – x)2*p(xi)i=1
i=k
Continuous Random Variables
• Continuous random variable can assume all real
number values within an interval (e.g., rainfall, pH)
• The probability distribution of a random
continuous variable is described by probability
density functions (pdf)
• A probability density function (pdf) is usually
represented by a graph or equation
• Again, there are two fundamental requirements for
a probability density function (pdf):
0)( xf
1)( dxxf
x
area=1
f(x)
µ
Probability Density Functions
• Theoretically, a continuous variable’s range can extend from negative infinity to infinity, e.g. the normal distribution:
• The tails of the normal distribution’s curve extend infinitely in each direction, but the value of f(x) approaches zero, getting closer and closer, but never reaching zero
x
area=1f(x)
• The probability of a continuous random variable X within an arbitrary interval is given by:
• Simply calculate the shaded shaded area if we know the density function, we could use calculus
x
f(x)
a b
b
adxxfbXap )()(
Probability Density Functions
• Fortunately, we do not need to solve the integral ourselves to practice statistics … instead, if we can match the f(x) up to some known distribution, we can use a table of probabilities that someone else has developed
• Tables A.2 through A.6 in the epilogue of the Rogerson text (pp. 214-221) give probability values for several distributions, including the normal distribution and some related distributions used by various inferential statistics
Probability Density Functions
• Suppose we are interested in computing the probability of a continuous random variable at a certain value of x (e.g. at d):
• As the interval from c to d becomes vanishingly narrow, the area below the curve within it becomes vanishingly small
x
f(x)a b
d• Can we find the probability of a value occurring at d? p(d) = ?
p(x)c
d
0 as c d
c
• No, p(d) = 0 … why? The reasons is: