ATMO 632: Week 1

               Review of Concepts in Probability & Statistics

                                      http://www.cgd.ucar.edu/~svn/atmo632/week1.htm

 

Reading Material

 

  1. Wilks: Chapters, 1, 2, 3.4-3.6, 4 (Binomial, Poisson, and Gaussian distributions), 5.1-5.2
  2. North: Chapter 1

 

 

Concepts

 

            Events, sample, population

                        Event: daytime high temperature above 100 F

                        Sample: The last 10 summers

                        Population: The last 1000 summers

 

            Dependent and independent events

                        Days when power consumption is very high

                        Days when the stock market is up

 

            Probability properties

                        Non-negative, sum to unity

                        Independent events

 

            Probability distribution function (PDF)

                        Mean, mode, median

Expectations & moments, moment generating function

 

Discrete distributions (Binomial, Poisson)

 

Continuous distributions (Gaussian, Student-t, Chi-square)

 

The difference between weather and climate

PDF of Lorenz system vs. North Atlantic Oscillation

                        Lorenz system: bimodal PDF

                        NAO: Unimodal PDF

 

Central Limit Theorem

           

Hypothesis testing

            Degrees of freedom

            z, t, and chi-square statistics                                          (Please Turn Over)

ATMO 632: Homework for week 1 (due Sep. 7)

 

  1. A “hundred-year flood” is a flood that occurs once every 100 years, on the average.
    1. What distribution would you use to model hundred-year floods on a year-to-year basis?
    2. Over a 100 year period, compute the probability that a hundred-year flood occurs (i) 0 times, (ii) exactly 1 time, and (iii) more than 1 time.
    3. Assume there are 100 river basins in the United States, each with the same probability of having a hundred-year flood. Compute the probability that in a given year, at least two river basins have a hundred-year flood.

 

  1. Google’s stock is trading at $100 today. A sophisticated computer model predicts that tomorrow, the stock will trade at one of the following prices, all with equal probability:

                 103, 101, 111, 99, 107, 109, 11, 104, 108

 

    1. Considering the above as “stock price events”, which type of event is more likely: stock price will be higher tomorrow, or lower?
    2.  If you own Google stock and want to sell it, should you sell today, or wait until tomorrow?

 

  1. (Wilks, Exercise 2.4) The effect of cloud seeding on suppression of damaging hail is being studied in your area, by randomly seeding or not seeding equal numbers of candidate storms. Suppose the probability of damaging hail from a seeded storm is 0.10, and the probability of damaging hail from an unseeded storm is 0.40. If one of the candidate storms has just produced damaging hail, what is the probability that it was seeded?

 

 

“Extra credit”

 

Write a program to solve the Lorenz system of equations

 

    dX/dt = s (Y – X)

    dY/dt = -XZ + rX – Y

    dZ/dt = XY – bZ

 

for parameter values s = 10, r = 28, b = 8/3. Starting from a random initial condition, integrate for a long time (i.e., until the trajectory has spanned the two lobes of the attractor several hundred times or more). Plot the (X,Y) trajectory and the probability distribution function (PDF) of the X coordinate. Repeat the same starting from an ensemble of 100 different randomly-chosen initial conditions, but plot the “ensemble-mean trajectory”, i.e., the trajectory defined by the average of the (X,Y) values of each of the 100 individual trajectories. Plot the PDF of the X coordinate for the ensemble-mean trajectory.