Transcript lecture_11_distribution_examplesx

Lecture 11
Introduction to
Probability - III
John Rundle Econophysics PHYS 250
Probability Distributions
• Q: Why should we care about probability
distributions? Why not just focus on the data?
• A: Outliers. We want to know how probable
are large market moves, so we can control our
exposure and risk
Probability Distributions: Gaussian
https://en.wikipedia.org/wiki/Normal_distribution
• In probability theory, the normal (or Gaussian) distribution is a very common
continuous probability distribution.
• Normal distributions are important in statistics and are often used in the natural
and social sciences to represent real-valued random variables whose
distributions are not known.
Density Function
Cumulative Distribution Function
Probability Distributions: Gaussian
https://en.wikipedia.org/wiki/Normal_distribution
• The normal distribution is useful because of the central limit theorem.
• In its most general form, under some conditions (which include finite
variance), it states that averages of random variables independently drawn
from independent distributions converge in distribution to the normal,
that is, become normally distributed when the number of random
variables is sufficiently large.
• Physical quantities that are expected to be the sum of many independent
processes (such as measurement errors) often have distributions that are
nearly normal.[3]
• Moreover, many results and methods (such as propagation of uncertainty
and least squares parameter fitting) can be derived analytically in explicit
form when the relevant variables are normally distributed.
Probability Distributions: Gaussian
https://en.wikipedia.org/wiki/Normal_distribution
Standard Normal Distribution
https://en.wikipedia.org/wiki/Normal_distribution
Gaussian Distribution: CDF
https://en.wikipedia.org/wiki/Normal_distribution
Gaussian Distribution: CDF
https://en.wikipedia.org/wiki/Normal_distribution
Confidence Intervals
https://en.wikipedia.org/wiki/Normal_distribution
Central Limit Theorem
https://en.wikipedia.org/wiki/Normal_distribution
Fourier Transform
https://en.wikipedia.org/wiki/Normal_distribution
Summary of Normal
Distribution
https://en.wikipedia.org/wiki/N
ormal_distribution
Log Normal Distribution
https://en.wikipedia.org/wiki/Log-normal_distribution
• In probability theory, a log-normal (or lognormal) distribution
is a continuous probability distribution of a random variable
whose logarithm is normally distributed.
• Thus, if the random variable is log-normally distributed, then
Y = ln (X) has a normal distribution.
• Likewise, if Y has a normal distribution, then X = exp(Y) has a
log-normal distribution.
• A random variable which is log-normally distributed takes only
positive real values.
• Whereas the Normal distribution describes a sum of random
variables, the LogNormal distribution describes a product of
random variables
Log Normal Distribution
https://en.wikipedia.org/wiki/Log-normal_distribution
Application of LogNormal Distribution
to Stock Prices
Because of compounding, the change in price after N
days (in %) will be a product of N factors.
However, in real markets, there are non-random
factors that lead to non-random (persistent) changes
in stock prices.
Summary of LogNormal
Distribution
https://en.wikipedia.org/wiki/L
og-normal_distribution
Student’s t-Distribution
https://en.wikipedia.org/wiki/Student's_t-distribution
• In probability and statistics, Student's t-distribution (or simply
the t-distribution) is any member of a family of continuous
probability distributions that arises when estimating the mean
of a normally distributed population in situations where the
sample size is small and population standard deviation is
unknown.
• Whereas a normal distribution describes a full population, tdistributions describe samples drawn from a full population
• Accordingly, the t-distribution for each sample size is
different, and the larger the sample, the more the distribution
resembles a normal distribution.
Student’s t-Distribution
https://en.wikipedia.org/wiki/Student's_t-distribution
Student’s t-Distribution
https://en.wikipedia.org/wiki/Student's_t-distribution
Lorentz (Cauchy) Distribution
https://en.wikipedia.org/wiki/Cauchy_distribution
• The Cauchy distribution, named after Augustin Cauchy, is a
continuous probability distribution. It is also known, especially
among physicists, as the Lorentz distribution (after Hendrik
Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or
Breit–Wigner distribution.
• The Cauchy distribution is often used in statistics as the canonical
example of a "pathological" distribution since both its mean and its
variance are undefined
• Its importance in physics is the result of it being the solution to the
differential equation describing forced resonance.
• In spectroscopy, it is the description of the shape of spectral lines
which are subject to homogeneous broadening in which all atoms
interact in the same way with the frequency range contained in the
line shape.
Lorentz (Cauchy) Distribution
https://en.wikipedia.org/wiki/Cauchy_distribution
Density Function
Cumulative Distribution Function
Lorentz Density Function
https://en.wikipedia.org/wiki/Cauchy_distribution
Lorentz Distribution: CDF
https://en.wikipedia.org/wiki/Cauchy_distribution
Lorentz Distribution: Characteristic Function
https://en.wikipedia.org/wiki/Cauchy_distribution
Summary of Lorentz
Distribution
https://en.wikipedia.org/wiki/
Cauchy_distribution
Exponential Distribution
https://en.wikipedia.org/wiki/Exponential_distribution
• In probability theory and statistics, the exponential
distribution is the probability distribution that describes the
time between events in a Poisson process, i.e. a process in
which events occur continuously and independently at a
constant average rate.
• It is a particular case of the gamma distribution. It is the
continuous analogue of the geometric distribution, and it has
the key property of being memoryless.
• In addition to being used for the analysis of Poisson processes,
it is found in various other contexts.
Exponential Distribution
https://en.wikipedia.org/wiki/Exponential_distribution
Exponential
Distribution
https://en.wikipedia.org/wiki/E
xponential_distribution
Poisson Distribution
https://en.wikipedia.org/wiki/Poisson_distribution
• In probability theory and statistics, the Poisson distribution is
a discrete probability distribution that expresses the
probability of a given number of events occurring in a fixed
interval of time and/or space if these events occur with a
known average rate and independently of the time since the
last event.
• The Poisson distribution can also be used for the number of
events in other specified intervals such as distance, area or
volume.
• The Poisson distribution is a 1-parameter distribution
• The time intervals between arrivals in a Poisson process are
exponentially distributed at the Poisson rate l
Poisson Distribution
https://en.wikipedia.org/wiki/Poisson_distribution
Poisson Distribution
https://en.wikipedia.org/wiki/Poisson_distribution
Poisson
Distribution
https://en.wikipedia.or
g/wiki/Poisson_distribu
tion
Binomial Distribution
https://en.wikipedia.org/wiki/Binomial_distribution
• In probability theory and statistics, the binomial distribution with
parameters n and p is the discrete probability distribution of the number
of successes in a sequence of n independent yes/no experiments, each of
which yields success with probability p.
• A success/failure experiment is also called a Bernoulli experiment or
Bernoulli trial
• When n = 1, the binomial distribution is a Bernoulli distribution.
• The binomial distribution is the basis for the popular binomial test of
statistical significance.
• The binomial distribution is frequently used to model the number of
successes in a sample of size n drawn with replacement from a population
of size N.
Binomial Distribution
https://en.wikipedia.org/wiki/Binomial_distribution
Binomial Distribution
https://en.wikipedia.org/wiki/Binomial_distribution
Binomial Distribution
https://en.wikipedia.org/wiki/Binomial_distribution
Weibull Distribution
https://en.wikipedia.org/wiki/Binomial_distribution
Weibull Distribution
https://en.wikipedia.org/wiki/Binomial_distribution
• If the quantity X is a "time-to-failure", the Weibull distribution
gives a distribution for which the failure rate is proportional to
a power of time
• A value of k < 1 indicates that the failure rate decreases over
time. This happens if there is significant "infant mortality", or
defective items failing early.
• A value of k = 1 indicates that the failure rate is constant over
time. This might suggest random external events are causing
mortality, or failure. The Weibull distribution reduces to an
exponential distribution;
• A value of k > 1 indicates that the failure rate increases with
time. This happens if there is an "aging" process, or parts that
are more likely to fail as time goes on.
Weibull Distribution
https://en.wikipedia.org/wiki/Bin
omial_distribution