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## Week 5

by: Claira Notetaker

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4

# Week 5 Q SCI 381

Claira Notetaker
UW
GPA 3.5

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Notes for Week 5
COURSE
Introduction to Probability and Statisitics
PROF.
Patrick C, Tobin
TYPE
Class Notes
PAGES
4
WORDS
CONCEPTS
Statistics
KARMA
25 ?

## Popular in Environmental Science

This 4 page Class Notes was uploaded by Claira Notetaker on Thursday February 4, 2016. The Class Notes belongs to Q SCI 381 at University of Washington taught by Patrick C, Tobin in Winter 2016. Since its upload, it has received 41 views. For similar materials see Introduction to Probability and Statisitics in Environmental Science at University of Washington.

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Date Created: 02/04/16
Midterm 2 2/4/16 3:17 PM Chapter 4: discrete Probability Distribution • sections 4.1-4.3 Random variable • (X) is a value observed from an outcome of an experiment that can be express numerically • EX: o Flipping a coin o Isn’t actually numeric but we can assign tails=1 and heads=2 and then we can observe the outcome Discrete Probability Distribution • There is a finite number of possible outcomes; expressed as an integer Discrete probability is a list of all of the possible outcomes that a random variable can assume with the corresponding probability of each outcome o The sum of the probability must equal 1 o The probability values have to be between 0 and 1 • The average/mean does not hold the correct weight in the typical sense o mean = (series) xP(x) § x = outcome of each test § P(x) = probability • Standard deviation= (variance) ^ ½ • Variance = (series) (x-u)^2 P(x) Binomial Distribution • Has a binary outcome (2 separate outcomes) o From a series of independent experiments or trials • P(x) = the probability of exactly x (frequency of the first outcome) in n (the total number of trials) • o n = number of independent trials o p = the probability of the first outcome o q = the probability of the second outcome § q = 1-p o x = the frequency of the first outcome in n trials § n—x = the frequency of the second outcome • mean = np • variance = (standard dev.)^2 = npq o standard deviation = (npq)^1/2 Geometric distribution • Considers the number of trials needed to achieve a desired outcome o Trials are independent Poisson distribution • Considers the probability of a given number of events occurring given the average rate of occurrence o Assuming events occur independently • P(x) = the probability of random variable x occurring in a set time period • o u = mean number of occurrences in the time period o x = the number of occurrences o e = an irrational number § 2.718 Chapter 5: Normal probability Distribution • section 5.1-5.4 Random Variable • Def: random variable, x is a value observed from an outcome of an experiment that can be expressed numerically • Has many possible outcomes: expressed as a real number Continuous probability distribution • Normal Gaussian Distribution • Many techniques (regression, correlation, analysis or variance) assume that the underlying distribution of data is normally distributed o Mean = 0 § Mean = median = mode o SD = 1 § 95% of data entries fall within 2SD moment statistics • using moments to significantly test for normality, g1 and g2 o 1 moment = mean nd o 2 moment = variance § V = sum of {(x-mean)^2/(n-1)} rd o 3 moment (g1) = skewness § sum of {(x-mean)^3)}/(n-1)(n-2)s(d^3) § theoretical value = 1.96 ú should be less than or equal too (absolute value) o 4 thmoment (g2) = kurtosis § sum of {(x-mean^4)}/(n-1)(n-2)(n-3)s(d^4) § theoretical value = 1.96 ú should be less than or equal too (absolute value) § focus on the “shoulders” of the graph ú low and fat curve: g2 < 0 ú normal: g2 = 0 ú tall and skinny: g2 > 0 z-scores • tells use how many standard deviation units a particular datum is form the mean • (x-mean)/SD • can use these markers to find the area under the curve at a given SD o can help us find the probability of a choice at a given data point o need a table which is in the book and on canvas Finding individual probability • us z-scores based on a sample set mean and standard deviation o z = (datum—mean) /SD • too find the probability fo the mean from a new sample set o use z-score based on the current estimate of the population mean and the standard error of the population mean (SEM) o z = (new mean—population mean) /SEM o SEM = SD/ (n^1/2) § SD = variance^1/2 Central limit theorem • The mean estimated from a sufficiently large number of samples will be approximately normally distributed, regardless of the underlying distribution Law of large numbers • The average of the results obtained from a large number of trials should approximate the expected value ***these are the 2 fundamental theorems of probability!

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