Lecture 2: Probability Distributions

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1 EAS31136/B9036: Statistics in Earth & Atmospheric Sciences Lecture 2: Probability Distributions Instructor: Prof. Johnny Luo

2 Dates Topic Reading (Based on the 2 nd Edition of Wilks book) Other Activity Aug 31 Introduction; Review of probability Wilks, Chap 2 Pre-test Sep 7 Matlab tutorial (optional) Sep 14 Review of probability; Probability Distribution 1 Wilks, Chap 2, 3 Sep 21 Probability Distribution 2 Wilks, Chap 3, 4 Sep 28 Hypothesis testing Wilks, Chap 5 Oct 5 Linear regression I Wilks Chap 6; von Storch 8-9 Oct 12 Linear regression II Wilks Chap 6; von Storch 8-9 Oct 19 Time series analysis I Wilks 8; von Storch Oct 26 Midterm; discussion of final project Nov 2 Time series analysis II Wilks 8; von Storch Project 1-page abstract due Univariate Statistics Nov 9 Nov 16 Principal Component Analysis & Empirical orthogonal functions I Principal Component Analysis & Empirical orthogonal functions II Wilks 11; von Storch 13 Wilks 11; von Storch 13 Project progress report due Multivariate Statistics Nov 30 Cluster analysis Wilks 14 Dec 7 Final project presentation

3 Probability Distribution 1. Definition of Terms 2. Some Empirical & Exploratory Data Analysis 3. Parametric Distribution I: Discrete Distributions 4. Parametric Distribution II: Continuous Distributions 5. Assessments of the Goodness of Fit

4 Ø Random Variable: when the value of a variable is the outcome of a statistical experiment (i.e., uncertain and dependent on chances), it is called a random variable. Ø Probability Distribution: Probability (remember: it means a ratio) assigned to values of a random variable. Ø Empirical Probability Distribution: just describe what s been observed (i.e., toss a coin 1000 times and you get probabilities for head & tail) an exploratory approach Ø Parametric Probability Distribution: summarize the observed probability distribution using particular mathematical forms (e.g., Gaussian distribution).

5 Probability Distribution 1. Definition of Terms 2. Some Empirical & Exploratory Data Analysis 3. Parametric Distribution I: Discrete Distributions 4. Parametric Distribution II: Continuous Distributions 5. Assessments of the Goodness of Fit

6 Describe data in n-quantiles 2-quantiles: Median 3-quantiles: Terciles 4-quantiles: Quartiles 100-quantiles: Percentiles How to calculate them? Step 1: rank the data in ascending (or descending) order; in matlab, the function for doing this is called sort. Step 2: find cutoff values for equal size subgroups

7 Box-and-whisker plots

8 Histograms Histograms of the Jan Max temperature in Ithaca. In Matlab, the function hist will plot histogram of a data array.

9 Cumulative Frequency Distribution Relative frequency for the probability (meaning a ratio) that an arbitrary future datum will not exceed the corresponding value on the horizontal axis.

10 Exploratory techniques for paired data Ø Scatterplot (In matlab, use plot ) Ø Pearson (ordinary) correlation (In matlab, use corrcoef )

11 Probability Distribution 1. Definition of Terms 2. Some Empirical & Exploratory Data Analysis 3. Parametric Distribution I: Discrete Distributions 4. Parametric Distribution II: Continuous Distributions 5. Assessments of the Goodness of Fit

12 Why bother with Parametric Distribution? Ø Empirical Probability Distribution: just describe what s been observed an exploratory approach Ø Parametric Probability Distribution: summarize the observed probability distribution using particular mathematical forms. Ø Mathematically more compact Ø More convenient for interpolation and extrapolation

13 Why bother with Parametric Distribution? Ø Empirical Probability Distribution: just describe what s been observed an exploratory approach Ø Parametric Probability Distribution: summarize the observed probability distribution using particular mathematical forms. q Binomial Distribution q Poisson Distribution q Gaussian Distribution q Gamma Distribution

14 Discrete Distribution I: Binomial Distribution Definition: A sequence of n independent yes/no (or head/tail) experiments. Usually we use 1 to represent yes and 0 for no. Random Variable (X): number of yes (or head) in a sequence of n trials. If you flip coin 3 times, what are possible values for X? X = 0, 1, 2, 3. Think-Pair-Share: which value has the largest probability and which one has the smallest?

15 For N=3, possible X = 0, 1, 2, 3. What are the probabilities of all these four possible outcomes? 0 (all 3 tails): 1/8 1 (1 head/2 tails): 3/8 2 (2 heads/1 tail): 3/8 3 (all 3 heads): 1/8

0 (all 3 tails): 1/8 1 (1 head/2 tails): 3/8 2 (2 heads/1tail): 3/8 3 (all 3 heads):

16 For N=3, possible X = 0, 1, 2, 3. What are the probabilities of all these four possible outcomes? 0 (all 3 tails): 1/8 1 (1 head/2 tails): 3/8 2 (2 heads/1tail): 3/8 3 (all 3 heads): 1/8 More Generally, Binomial distribution is a discrete parametric distribution with two parameters: 1) N, 2) p (i.e., probability for head at each trial)

trials (google combination and permutation ) 3 choose 2: red, green and blue

17 factorial (3x2x1)/(2x1x1) = 3 5 choose 3: How many combinations do we have? N choose x : different ways of distributing x successes in a sequence of N trials (google combination and permutation ) 3 choose 2: red, green and blue balls, choose 2 at a time, how many combinations do we have? (5x4x3x2x1)/(3x2x1x2x1) = 10

18 = 0.5 x (1-0.5) N-x = 0.5 N = = 1/8 For N=3, possible X = 0, 1, 2, 3. 0 (all 3 tails): 3!/(0!3!)(1/8) = 1/8 1 (1 head/2 tails): 3!/(1!2!)(1/8)= 3/8 2 (2 heads/1tail): 3!/(2!1!)(1/8)= 3/8 3 (all 3 heads): 3!/(3!0!)(1/8)= 1/8

19 Combination of x successes out of N trials Pr{success for x times} Pr{fail for N-x times} Independence & Multiplicative Law of Probability Two events are independent if the occurrence or nonoccurrence of one does not affect the probability of the other.

20 Combination of x successes out of N trials Pr{success for x times} Pr{fail for N-x times} One head: [3!/(1!2!)] (1/2) 1 (1-1/2) 2 = 3/8 3 combinations of getting 1 head + 2 tail Head once (1/2) Tail twice (1/4)

21 Application in Earth Sciences 220-yr record Step 1: Find the two parameters in the binomial distribution Q1: Compute the probability of the lake freezing next winter (or in any single winter in the future) x = 1 (1 freeze yr) N = 1 (only one future year) p = 10/220 = Step 2: Pr{X=1} = (1!)/(1! 0!) (0.045)( ) 1-1 = This is trivial!

22 Application in Earth Sciences 220-yr record Step 1: Find the two parameters in the binomial distribution Q2: Compute the probability of the lake freezing once in 10 years. x = 1 (1 freeze yr) N = 10 future years p = 10/220 = Step 2: Pr{X=1} = (10!)/(1! 9!) (0.045)( ) 10-1 = 0.30

23 Application in Earth Sciences 220-yr record Step 1: Look for the complement event x = 0 (no freezing at all in 10 years) x = 0 (no freeze yr); N = 10; p = 10/220 = Q3: Compute the probability of the lake freezing at least once in 10 years. Step 2: Pr{X=0} = (10!)/(0! 10!) (0.045) 0 ( ) 10 = 0.63 Pr{X=1} = = 0.37

24 A Few Insights concerning Binomial Distribution Two conditions: q The probability of the event doesn t change from trial to trial. q The outcomes of the N trials are mutually independent Possible violations: q Thunderstorm or lightning occurrence: during summer time over land, thunderstorms favors afternoon q Six-hourly precipitation occurrence: the same weather system could last longer than 6 hours, so rain now probably means a large chance of rain 6 hours after.

25 Source: wikipedia

Lecture 4: Statistical Hypothesis Testing

EAS31136/B9036: Statistics in Earth & Atmospheric Sciences Lecture 4: Statistical Hypothesis Testing Instructor: Prof. Johnny Luo www.sci.ccny.cuny.edu/~luo Dates Topic Reading (Based on the 2 nd Edition