Download Slides 3: Probablity (PDF, 153 KB)

Stochastic Models
Probability
Walt Pohl
Universität Zürich
Department of Business Administration
February 28, 2013
What is Probability?
Mathematically, it’s just a list of axioms:
1
Probabilities between 0 and 1.
2
Probabilities of all possibilities add up to 1.
3
If two things, A and B can’t both happen at the
same time, then probability of A or B equals
probability of A + probability of B.
But when do these axioms apply?
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Stochastic Models
February 28, 2013
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The Three Justifications of Probability
1
2
3
Subjective probability
Objective probability
Market probability
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Stochastic Models
February 28, 2013
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Objective Probability
Objective probability – a probability statement is an
objective fact about the world.
If you flip a fair coin a large number of times, it will
come up heads 50% of the time.
But what about events that only happen once?
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Stochastic Models
February 28, 2013
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Subjective Probability
Subjective probability – a probability statement is a
subjective belief how likely something is.
Usually formatted as betting odds. You believe that a
coin is fair if you’re willing to bet 0.5 to win 1 that it
comes up heads.
In practice, probability is frequently subjective by
necessity.
Walt Pohl (UZH QBA)
Stochastic Models
February 28, 2013
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Market Probability
Market probability – the betting odds assign in the
marketplace.
In 2010, SkyBet gave Switzerland a 55% chance of
beating Honduras in the World Cup. Did SkyBet believe
that Switerland has a 55% chance of winning? Probably
not.
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Stochastic Models
February 28, 2013
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Probability Distributions
We usually work with real-valued probability
distributions: for each range [a, b], we can calculate
P(a ≤ x ≤ b).
Sufficient to know the cumulative distribution
function (cdf):
F (a) = P(x ≤ a)
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Stochastic Models
February 28, 2013
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Discrete Probability Distributions
Can only take on a finite number of values:
a1 , . . . , an .
Can be given by probability mass function,
f (a) = P(x = a).
CDF is
F (a) =
X
f (ai ).
ai <a
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Stochastic Models
February 28, 2013
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Continuous Probability Distributions
Can take a continuous range of values.
Can be given by probability density function,
f (a).
CDF is
Z a
f (x)dx.
−∞
Probability of any single real number happening is
zero.
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Stochastic Models
February 28, 2013
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Other Possibilities
Discrete and continuous probability distributions are not
the only possibilities.
Let X be normally distributed, and define a new random
variable as min(X , 0). Then this is a discrete random
variable 50% of the time (exactly equal to zero), and a
continuous random variable 50% of the time.
There are more exotic possibilities, related to fractals.
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Stochastic Models
February 28, 2013
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Working with Probability Distributions
We typically want to know certain things.
Moments:
Mean
Variance
Quantiles:
Median
Extremes: 5%, 95%
We get these values either analytically or by Monte Carlo
simulation.
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Stochastic Models
February 28, 2013
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Aside on Quantiles
For continuous distributions, quantiles and the CDF are
closely related. The q-th quantile is a such that
P(x ≤ a) = q.
If F is the CDF, then F −1 (q) is the q-th quantile.
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Stochastic Models
February 28, 2013
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Analytically
For discrete distributions, you can compute everything by
brute force. For continuous distributions, you can do it
only certain cases.
Most important case: normal distribution.
√
1
2πσ 2
e
−(x−µ)2
2σ 2
We can show the mean is µ and the variance is σ 2 .
Let Φ be the CDF. Then Φ−1 (q) is the q-th quantile.
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Stochastic Models
February 28, 2013
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Monte Carlo Simulation
We simulate draws from the distribution on a computer.
We then work with these statistically – to compute the
mean, standard deviation, or quantiles, we use the
sample quantities computed from the random draws.
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Stochastic Models
February 28, 2013
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Why Does Monte Carlo Work?
Monte Carlo simulation generates a simulated CDF,
which approximate the true CDF.
Glivenko-Cavelli Theorem: As the number of draws goes
to infinity, the simulated CDF converges to the true CDF.
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Stochastic Models
February 28, 2013
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