1. Introduction to Probability - III John Rundle Econophysics PHYS 250
Lecture 11
Introduction to
Probability - III
John Rundle Econophysics PHYS 250

2. 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

3. Cumulative Distribution Function
Probability Distributions: Gaussian
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

4. Probability Distributions: Gaussian https://en. wikipedia
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.

5. Probability Distributions: Gaussian https://en. wikipedia

6. Standard Normal Distribution https://en. wikipedia

7. Gaussian Distribution: CDF https://en. wikipedia

8. Gaussian Distribution: CDF https://en. wikipedia

9. Confidence Intervals https://en.wikipedia.org/wiki/Normal_distribution

10. Central Limit Theorem https://en. wikipedia

11. Fourier Transform https://en.wikipedia.org/wiki/Normal_distribution

12. Summary of Normal Distribution

13. Log Normal Distribution https://en. wikipedia
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

14. Log Normal Distribution https://en. wikipedia

15. 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.

16. Summary of LogNormal Distribution

17. Student’s t-Distribution https://en. wikipedia
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, t-distributions 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.

18. Student’s t-Distribution https://en. wikipedia

19. Student’s t-Distribution https://en. wikipedia

20. Lorentz (Cauchy) Distribution https://en. wikipedia
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.

21. Cumulative Distribution Function
Lorentz (Cauchy) Distribution
Density Function
Cumulative Distribution Function

22. Lorentz Density Function https://en. wikipedia

23. Lorentz Distribution: CDF https://en. wikipedia

24. Lorentz Distribution: Characteristic Function https://en. wikipedia

25. Summary of Lorentz Distribution

26. Exponential Distribution https://en. wikipedia
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.

27. Exponential Distribution https://en. wikipedia

28. Exponential Distribution https://en. wikipedia

30. Poisson Distribution https://en. wikipedia
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

31. Poisson Distribution https://en. wikipedia

32. Poisson Distribution https://en. wikipedia

33. Poisson Distribution https://en. wikipedia

35. Binomial Distribution https://en. wikipedia
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.

36. Binomial Distribution https://en. wikipedia

37. Binomial Distribution https://en. wikipedia

38. Binomial Distribution https://en. wikipedia

39. Weibull Distribution https://en. wikipedia

40. Weibull Distribution https://en. wikipedia
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.

41. Weibull Distribution https://en. wikipedia