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EDSA: Lecture 2
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Further info on probability distributions
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We typically use probability distributions because they work (i.e., they fit lots of data in real world)
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Remember Discrete vs Random
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Bernoulli Random Variables
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Variable with only two possible outcomes
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Success (S) or death (D) etc.
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Failure (F) or alive (A) etc.
Examples
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Heads or Tails
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Newborn baby sex (male or female)…ok not a good example
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Live or die (although funny story about this…)
Lets define binomial random variable X to be the # of successful results in n independent
Bernoulli trials (i.e. replicated)
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Let p = probability of successes
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Let q = 1-p = probability of failure
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If n=1, then binomial random variable X is = to Bernoulli trial
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What is the probability of obtaining x successes in n trials?
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Example
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What is the probability of obtaining 2 tails from a coin that was tossed 5 times?
5
P(TTHHH) = (1/2) = 1/32
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But there are more possibilities:
TTHHH THTHH THHTH THHHT
HTTHH HTHTH HTHHT
HHTTH HHTHT
HHHTT
P(2 tails) = 10 × 1/32 = 10/32
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In general, if trials result in a series of success and failures,
FFSFFFFSFSFSSFFFFFSF…
Then the probability of x successes in that order is
P(x)
=qqpq
n–x
=p q
x
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However, if order is not important, then
where
is the number of ways to obtain x
successes in n trials, and i! = i  (i – 1)  (i – 2)  …  2  1
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Where X ~ Bin(n,p)
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P(X) =
n!
x!(n-x)!
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where P(X) is the probability of exactly x successes, n is the # of trials, p is the probability
of success in any one trial
x
n-x
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The probability of obtaining x successes and (n-x) failures is p (1-p)
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n!/(x!(n-x)!) is the number of ways to obtain x success in n trials, accounts for controls double
counting, e.g. (1,0)=(0,1)
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Example
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We shall use the tails example
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We will generate X ~ Bin (11, 0.5)
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Now lets play in excel.
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Muck with sample size and n.
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What happens to the distributions when p ≠ 0.5
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Lets do this in excel.
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(8, 11) (1, 11) (9, 11) (3, 11) (5, 11)
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We have now purposefully skewed the data:
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Skew: describes distribution asymmetry
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Two peaks
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Leptukurtic: more observations (scores) in tails than predicted
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Platykurtic: less observations (scores) in tails than predicted
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Biology deals with tail probabilities
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What is the probability of obtaining 9 or 10 tails out of 19 coin flips?
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Answer = 17.6%
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All P-values for statistical tests are tail probabilities
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Other types of distribution
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Poisson -- # of instances of an event recorded in a sample of fixed intervals or areas.
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Usually distribution of rare events or counts
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Poisson distribution is applied where random events in space or time are expected to occur
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Deviation from Poisson distribution may indicate some degree of non-randomness in the events
under study
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Investigation of cause may be of interest
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X ~ Poisson(λ)
P(x) = λ
x
x!
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where λ is the average value of the # of occurrences of the event in each sample
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e is a constant, the base of the natural logarithm (~2.71828)
If the average # of seedlings found in a quadrat is 0.75, what are the chances a quadrat will have
4 seedlings?
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4
P(4 seedlings) = (0.75 x e
-0.75
)/4! = 0.0062
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Poisson Distributions
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Example
Emission of -particles
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Observed vs Expected Poisson
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A unweighted arithmetic mean can be misleading
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e.g. Binomial variable that can take on values of 0 and 50 with probabilities of 0.1 and
0.9, respectively
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Arithmetic mean = (0 + 50)/2 = 25
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But, the most probably value is 50
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Arithmetic mean is not be useful for skewed distributions
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Expected value of a discrete random variable X that can take on values a1
…an, with probabilities p1….pn, respectively, and accurately describes the
average or central tendency of the distribution
E(X) = ∑aipi = a1p1 + a2p2 +...+ anpn
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Limits of Central Tendencies
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Averages, or central tendencies, give no insight into spread, or variation. Need both.
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e.g. Binomial variable that can take on -10 or 10 vs one that can take on -1000 or +1000,
each with a p=0.5
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Both have E(X)=0, but in neither distribution will 0 ever be generated
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The observed values in the first case are closer to E(X) than in the second
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Variance of a random variable =
2
2
2
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σ (X) = E[X – E(X)] = ∑ pi (ai - ∑ aipi)
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E(X) is the expected value of X, the ai’s are the different possible values of X, each of
which occurs with probability pi
Interpretation
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We calculate E(X), subtract this from X, and square this difference
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Because there are many possible values of X, we repeat this process for each value
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Each squared deviate is weighted by its probability of occurrence, pi, and then they are
all summed
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Binomial vs Poisson
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Binomial depends on both n and p and Poisson depends on only λ
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Binomial is always bounded by 0 and n, whereas the right-hand tail of the Poisson is not bounded
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E(X) = σ (X) for Poisson whereas E(X) typically does not equal σ (X) for a binomial
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Normal Distribution
Length of Fish
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A sample of rock cod in Monterey Bay suggests that the mean length of these fish is  = 40 in.
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and  = 4 in.
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Assume that the length of rock cod is a normal random variable
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If we catch one of these fish in Monterey Bay,
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What is the probability that it will be at least 41 in. long?
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That it will be no more than 42 in. long?
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That its length will be between 36 and 39 inches?
2