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**STATISTICS Sampling and Sampling Distributions**

Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

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Random sample Let the random variables X1, X2, …, Xn have a joint density that factors as follows: where is the common density of each Xi . Then (X1, X2, …, Xn) is defined to be a random sample of size n from a population with density 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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If X1, X2, …, Xn is a random sample of size n from , then X1, X2, …, Xn are stochastically independent. Histogram -- A frequency (or relative frequency) plot of observed data is called a frequency histogram (or relative frequency histogram). 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Frequency Histogram 3/27/2017**

Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Cumulative frequency 3/27/2017**

Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Relative cumulative frequency**

3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Statistic A statistic is a function of observable random variables, which is itself an observable random variable and does not contain any unknown parameters. A statistic must be observable because we intend to use it to make inferences about the density functions of the random variables. 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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For example, if a random variable has a probability density function where and are unknown, then is not a statistic. If a statistic is not observable, then it can not be used to inference the parameters of the density function. 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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An observation of random sample of size n can be regarded as n independent observations of a random variable. 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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One of the central problems in statistics is to find suitable statistics to represent parameters of the probability distribution function of a random variable. Sample Statistics Population Parameters Observable Unknown 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Sample moments Let X1, X2, …, Xn be a random sample from the density Then the rth sample moment about 0 is defined as 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**In particular, if r = 1, we have the sample mean ; that is,**

Also, the rth sample moment about the sample mean is defined as 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Theorem – Let X1, X2, …, Xn be a random sample from the density**

Theorem – Let X1, X2, …, Xn be a random sample from the density The expected value of the rth sample moment about 0 is equal to the rth population moment; i.e., Also, 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Special case: r=1 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Sample statistics Let X1, X2, …, Xn be a random sample from the distribution of a random variable X. Sample mean and sample variance of the distribution are respectively defined to be 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Estimating the mean Given a random sample from a probability density function f(．) with unknown mean μ and finite variance σ2, we want to estimate the mean using the random sample. Using only a finite number of values of X (a random sample of size n), can any reliable inferences be made about E(X), the average of an infinite number of values of X? Will the estimate be more reliable if the size of the random sample is larger? 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**R-program demonstration**

3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Mean of sample means w.r.t. sample size**

3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Mean of sample standard deviations w.r.t. sample size**

3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Standard deviation of sample means w.r.t. sample size**

What is the theoretical basis? Y=f(x)=? 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Histograms of sample mean and standard deviation ns=30**

3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Histograms of sample mean and standard deviation ns=5000**

3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Weak Law of Large Numbers (WLLN)**

Let f(．) be a density with mean μ and variance σ2, and let be the sample mean of a random sample of size n from f(．). Let ε and δ be any two specified numbers satisfying ε>0 and 0<δ<1. If n is any integer greater than , then 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Recall the theorem 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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(Example) Suppose that some distribution with an unknown mean has its variance equal to 1. How large a random sample must be taken such that the probability will be at least 0.95 that the sample mean will lie within 0.5 of the population mean? 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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(Example) How large a random sample must be taken in order that you are 99% certain that is within 0.5σ of μ? 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Raingauge network design**

Assuming there are already some raingauge stations in a catchment, and we are interested in determining the optimal number of stations that should exist to achieve a desired accuracy in the estimation of mean rainfall. Two approaches (1) Standard deviation of the sample mean should not exceed a certain portion of the population mean. (2) 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Criterion 1 Standard deviation of the sample mean should not exceed a certain portion of the population mean. 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**From the weak law of large numbers,**

Criterion 2 From the weak law of large numbers, What assumptions have we made for such approaches of network design ? What are the practical considerations in monitoring network design? Data independence 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**The Central Limit Theorem**

Let f(．) be a density with mean μ and finite variance σ2. Let be the sample mean of a random sample of size n from f(．). Then approaches the standard normal distribution as n approaches infinity. 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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The importance of the CLT is the fact that the mean of a random sample from any distribution with finite variance σ2 and mean μ is approximately distributed as a normal random variable with mean μ and variance 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**R-program demonstration - Central Limit Theorem**

3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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n=2 n=10 n=25 n=50 n=100 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Sampling distributions**

Given random samples of certain probability densities, we often are interested in knowing the probability densities of sampling statistics. Poisson distribution Exponential distribution Normal distribution Chi-square distribution Standard normal and chi-square distributions Student’s t-distribution 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Poisson distribution 3/27/2017**

Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Exponential distribution**

3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Normal distribution 3/27/2017**

Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Chi-square distribution**

3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Standard normal and chi-square distributions**

3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**Student’s t-distribution**

Student’s t distribution with k degrees of freedom 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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**The "student's" distribution was published in 1908 by W. S. Gosset**

The "student's" distribution was published in 1908 by W. S. Gosset. Gosset, however, was employed at a brewery that forbade the publication of research by its staff members. To circumvent this restriction, Gosset used the name "Student", and consequently the distribution was named "Student t-distribution. 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Order statistics 3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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3/27/2017 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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