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Probability Distributionsdistributions PartPart 1:1: ProbabilityProbability DistributionsDistributions Probability Distribution Normal Distribution Student-t Distribution Chi Square Distribution F Distribution Significance Tests ESS210B Prof. Jin-Yi Yu PurposesPurposes ofof DataData AnalysisAnalysis True Distributions or Relationships in the Earths System Organizing Data Sampling Find Relationships among Data Weather Forecasts, Test Significance of the Results Physical Understanding, …. ESS210B Prof. Jin-Yi Yu ParametersParameters andand StatisticsStatistics Parameters: Numbers that describe a population. For example, the population mean (µ)and standard deviation (σ). Statistics: Numbers that are calculated from a sample. A given population has only one value of a particular parameter, but a particular statistic calculated from different samples of the population has values that are generally different, both from each other, and from the parameter that the statistics is designed to estimate. The science of statistics is concerned with how to draw reliable, quantifiable inferences from statistics about parameters. (from Middleton 2000) ESS210B Prof. Jin-Yi Yu VariablesVariables andand SamplesSamples Random Variable: A variable whose values occur at random, following a probability distribution. Observation: When the random variable actually attains a value, that value is called an observation (of the variable). Sample: A collection of several observations is called sample. If the observations are generated in a random fashion with no bias, that sample is known as a random sample. By observing the distribution of values in a random sample, we can draw conclusions about the underlying probability distribution. ESS210B Prof. Jin-Yi Yu ProbabilityProbability DistributionDistribution continuous probability distribution discrete probability distribution PDFPDF The pattern of probabilities for a set of events is called a probability distribution. (1) The probability of each event or combinations of events must range from 0 to 1. (2) The sum of the probability of all possible events must be equal too 1. ESS210B Prof. Jin-Yi Yu ExampleExample If you throw a die, there are six possible outcomes: the numbers 1, 2, 3, 4, 5 or 6. This is an example of a random variable (the dice value), a variable whose possible values occur at random. When the random variable actually attains a value (such as when the dice is actually thrown) that value is called an observation. If you throw the die 10 times, then you have a random sample which consists of 10 observations. ESS210B Prof. Jin-Yi Yu ProbabilityProbability DensityDensity FunctionFunction P = the probability that a randomly selected value of a variable X falls between a and b. f(x) = the probability density function. The probability function has to be integrated over distinct limits to obtain a probability. The probability for X to have a particular value is ZERO. Two important properties of the probability density function: (1) f(x) ≥ 0 for all x within the domain of f. (2) ESS210B Prof. Jin-Yi Yu CumulativeCumulative DistributionDistribution FunctionFunction The cumulative distribution function F(x) is defined as the probability that a variable assumes a value less than x. The cumulative distribution function is often used to assist in calculating probability (will show later). The following relation between F and P is essential for probability calculation: ESS210B Prof. Jin-Yi Yu NormalNormal DistributionDistribution f: probability density function mean µ: mean of the population σ: standard deviation of the population The normal distribution is one of the most important distribution in geophysics. Most geophysical variables (such as wind, temperature, pressure, etc.) are distributed normally about their means. ESS210B Prof. Jin-Yi Yu StandardStandard NormalNormal DistributionDistribution The standard normal distribution has a mean of 0 and a standard deviation of 1. This probability distribution is particularly useful as it can represent any normal distribution, whatever its mean and standard deviation. Using the following transformation, a normal distribution of variable X can be converted to the standard normal distribution of variable Z: Z = ( X - µ ) / σ ESS210B Prof. Jin-Yi Yu TransformationsTransformations It can be shown that any frequency function can be transformed in to a frequency function of given form by a suitable transformation or functional relationship. For example, the original data follows some complicated skewed distribution, we may want to transform this distribution into a known distribution (such as the normal distribution) whose theory and property are well known. Most geoscience variables are distributed normally about their mean or can be transformed in such a way that they become normally distributed. The normal distribution is, therefore, one of the most important distribution in geoscience data analysis. ESS210B Prof. Jin-Yi Yu HowHow toto UseUse StandardStandard NormalNormal DistributionDistribution Example 1: What is the probability that a value of Z is greater than 0.75? Answer: P(Z≥0.75) = 1-P(Z≤0.75)=1-0.7734=0.2266 ESS210B Prof. Jin-Yi Yu AnotherAnother ExampleExample negative value Example 2: What is the probability that Z lies between the limits Z1=-0.60 and Z2=0.75? Answer: P(Z ≤-0.60) = 1 − P(Z < 0.60) Î P(Z >-0.60)=1−0.7257=0.2743 P(-0.60 ≤Z ≤0.75) = P(Z ≤0.75) − P(Z ≤-0.60) = 0.7734 − 0.2743 ESS210B = 0.4991 Prof. Jin-Yi Yu ExampleExample 11 Given a normal distribution with µ = 50 and σ = 10, find the probability that X assumes a value between 45 and 62. The Z values corresponding to x1 = 45 and x2 = 62 are Therefore, P(45 < X < 62) = P(-0.5 < Z < 1.2) = P(Z <1.2) – P(Z<-0.5) = 0.8849 – 0.3085 = 0.5764 ESS210B Prof. Jin-Yi Yu ExampleExample 22 An electrical firm manufactures light bulbs that have a length of life that is normally distributed with mean equal to 800 hours and a standard deviation of 40 hours. Find the possibility that a bulb burns between 778 and 834 hours. The Z values corresponding to x1 = 778 and x2 = 834 are z1 = (778 – 800) / 40 = -0.55 z2 = (834 – 800) / 40 =0.85 Therefore, P(778 < X < 834) = P(-0.55 < Z < 0.85) = P(Z <0.85) – P(Z<-0.55) = 0.8849 – 0.2912 = 0.5111 ESS210B Prof. Jin-Yi Yu ProbabilityProbability ofof NormalNormal DistributionDistribution ESS210B Prof. Jin-Yi Yu ProbabilityProbability ofof NormalNormal DistributionDistribution There is only a 4.55% probability that a normally distributed variable will fall more than 2 standard deviations away from its mean. This is the two-tailed probability. The probability that a normal variable will exceed its mean by more then 2 σ is only half of that, 2.275%. ESS210B Prof. Jin-Yi Yu ApplicationApplication inin OperationalOperational ClimatologyClimatology 0.53 -1.28 -0.53 Warmer than 9 out of 10 years 1.28 This figure shows average temperatures for January 1989 over the US, expressed as quantiles of the local Normal distributions. Different values of µ and σ have been estimated for each location. ESS210B Prof. Jin-Yi Yu HowHow toto EstimateEstimate µµ andand σσ In order to use the normal distribution, we need to know the mean and standard deviation of the population ÎBut they are impossible to know in most geoscience applications, because most geoscience populations are infinite. ÎWe have to estimate µ and σ from samples. and ESS210B Prof. Jin-Yi Yu SamplingSampling DistributionDistribution Is the sample mean close to the population mean? To answer this question, we need to know the probability distribution of the sample mean. We can obtain the distribution by repeatedly drawing samples from a population and find out the frequency distribution. ESS210B Prof. Jin-Yi Yu ExampleExample If we repeatedly draw N observations from a standard normal distribution (µ=0 and σ=1) and calculate the sample mean, what is the distribution of the sample mean? ESS210B (from Berk & Carey’s book) Prof. Jin-Yi Yu DistributionDistribution ofof SampleSample AverageAverage What did we learn from this example? ÎIf a sample is composed of N random observations from a normal distribution with mean µ and standard deviation σ, then the distribution of the sample average will be a normal distribution with mean µ but standard deviation σ/sqrt(N). ESS210B Prof. Jin-Yi Yu StandardStandard ErrorError The standard deviation of the probability distribution of the sample mean (X) is also referred as the “standard error” of X. ESS210B Prof. Jin-Yi Yu DistributionDistribution ofof SampleSample MeansMeans One example of how the normal distribution can be used for data that are “non-normally” distributed is to determine the distribution of sample means. Is the sample mean close to the population mean? To answer this question, we need to know the probability distribution of the sample mean. We can obtain the distribution by repeatedly drawing samples from a population and find out the frequency distribution. ESS210B Prof. Jin-Yi Yu AnAn ExampleExample ofof NormalNormal DistributionDistribution 1000 random numbers =2 N N = 10 non-normal distribution If we repeatedly draw a sample of N values from a population and calculate the mean of that sample, the distribution of sample means tends to be in normal distribution. ESS210B (Figure from Panofsky and Brier 1968) Prof. Jin-Yi Yu CentralCentral LimitLimit TheoryTheory In the limit, as the sample size becomes large, the sum (or the mean) of a set of independent measurements will have a normal distribution, irrespective of the distribution of the raw data. If there is a population which has a mean of µand a standard deviation of σ, then the distribution of its sampling means will have: µ µ (1) a mean x = , and σ σ 1/2 (2) a standard deviation x = /(N) = standard error of the mean ESS210B Prof. Jin-Yi Yu SampleSample SizeSize (S-σ)/σ * 100 N=30 The bigger the sample size N, the more accurate the estimated mean and standard deviation are to the true values.
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