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# Statistical Methods I STA 2023

University of Central Florida

GPA 3.67

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This 2 page Class Notes was uploaded by Dimitri Torphy on Thursday October 22, 2015. The Class Notes belongs to STA 2023 at University of Central Florida taught by Chung-Ching Wang in Fall. Since its upload, it has received 54 views. For similar materials see /class/227639/sta-2023-university-of-central-florida in Statistics at University of Central Florida.

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Date Created: 10/22/15

W A random variable is a variable that assumes values associated with the random outcomes of a random experiment where one and only one numerical values is assigned to each sample point The distance between you home and UCF is between 0 and 100 miles that is an interval ie the distance between your home and UCF is a continuous random variable The number of heads in coin tossing experiment is a count ie the number of heads in a coin tossing experiment is a discrete random variable Usually we can use the following four steps to complete a probability table Step 1 Find out the variable of interest Step 2 List all the sample points in the sample space Step 3 List all the possible values of this random variable Step 4 Assign the probabilities to all the possible values The probability distribution of a discrete random variable is a graph a table or a formula that specifies the probability associated with each possible value the random can assume The probability distribution should not include values that have zero probabilities Thus the probability of any value of a random variable is among 0 and 1 and the sum of the probabilities of all possible values of a random variable is equal to one Population Mean u Z xpx Ex Population Variance 02 Exu2 Zxu2x px Population Standard Deviation o hr 2 Binomial Random Variable characteristics First they consist of n identical and independent trials Second there are only two possible outcomes denoted by S and F on each trail Third the possibility of each outcome remains unchanged from trial to trial that is the probability ofS is p and probability of F is q1p Fourth we are interested in the random variable x represented the number ofS happened in n trails n is a xed number Therefore it is worth to develop a special probability model to deal with this kind of random variables Any random variable that has these four characteristics is called binomial random variable and can be dealt by using this special probability model Suppose thatX is a binomial random variable The probability of success on any single trial is p and there are n trials in this random experiment The probability density function ofX is PXx px pxqquot39xi x 012n p the probability of success on any single trial n total number of trials q 1 39P x number ofsuccesses in n trials Let u and o be the mean and standard deviation of the binomial random variable X Instead of using the expectation summation rules to calculate it and 0 we can nd it and o easily using the formulas u hp 02 npq np1pand 0 n q lnp1 p Poisson probability model The probability density function of a Poisson random variable is px A 1 Both the mean and the variance of a Poisson random variable equals to A ie u A and o2 A Collection of De nitions Random Variable A random variable is a rule that assigns one and only one numerical value to each sample point in a random experiment Discrete Random Variable Discrete random variable is one kind of random variable that can assume values on countable number of points Continuous Random Variable Continuous random variable is one kind of random variable that can assume values in one or more intervals Probability Distribution The probability distribution of a discrete random variable is a graph a table or a formula that speci es the probability associated with each possible value the random can assume Expectation of a Discrete Random Variable The expectation of a discrete random variable is the population mean of this random variable We can use the following formula to compute the expectation of a discrete random variable uprxEx Variance of Discrete Random Variable The variance of a discrete random variable is 02 Exu2 Zxu2px Standard Deviation of Discrete Random Variable The standard deviation of a discrete random variable is equal to the square root of the variance of this random variable ie 0 W Binomial Distribution The probability density function of a binomial random variable is PXx px C LpxqHJ where p the probability of success on any single trial n total number of trials x number ofsuccesses in n trials q 1 P The mean ofa binomial random variable is np ie u np The variance of a binomial random variable is npq ie o2 npq np1p x012 x 7A Poisson Random Variable The probability density function of a Poisson random variable is px A x012 Both the mean and the variance of a Poisson random variable equals to A ie u A and 02 A Hypergeometric Random Variable The probability density function of a Hypergeometric random variable is px xmax0nNrminrn Where N total number of elements in the population r the number ofsuccesses in the N elements n the number of elements drawn x the number ofsuccesses is drawn in the n elements The mean ofa Hypergeometric random variable is u and the variance of a Hypergeometric random variable is 02 W Chapter 5 Uniform distribution Continuous random variables that appear to have equal likely outcomes over their entire range of possible values possess a uniform distribution Suppose that the random variable x can assume values only in interval c lt x lt d Then the probability density function ofx is px L if div csxsd 2 d 2 C ando C respectively 12 12 Normal distribution Some reasons for the popularity of the normal distribution are as follows 1 The distributions of many random variables such as the height ofa group ofstudents the length of ears of corns the errors made in measuring a person39s blood pressure are approximately normally distributed 2 The normal distribution is relatively easy to work with mathematically Many results based on normal distribution may hold well enough for practical usage when samples come from nonnormal populations included populations of discrete random variab es 3 Some measurements do not have a normal distribution but a simple transformation of the original scale of the measurement may induce approximately normality 4 Even if the distribution in the original population is far from normal the distribution of sample means tends to become normally distributed if the sample sizes are suf ciently large 2d The expectation the variance and the standard deviation ofx are u dzi 039 x 2 The probability density function of a normal random variable with mean u and standard deviation 0 is px e 2er2 if ltxlt Normal distribution has several very good properties First the normal distribution is symmetric Second the normal distribution has an unique mode Second the normal distribution has a unique mode Third the population mean the population median and population mode of a normal random are overlap The standard normal random variable is a normal random variable with mean zero and standard deviation one lfx is a normal random variable with mean u and standard deviation of o z 0quot is a standard normal random variable Steps to nd probabilities associate with a normal random variable from a normal prob table 1 Draw the normal curve and shade the area corresponding to the probability for which you want to nd 2 Convert the x values to standard normal random variable val ues z using the formula z x u l s 3 Use the table IV in Appendix A or the table on the back of front cover page to nd the area corresponding to the z values Exponential Distribution good probability model to study the amount of time or distance between the occurrences of 2 random events Chapter 6 Parameter the uknown numerical values that is used to describe the properties of a population Sample statistics the computed numerical values from the measurements in a samp e Sampling error error results from using a sample instead of census the population to estimate a population quantity Sampling distribution sample stats are vary from one sample to another Therefore there is a distribution function associated with each sample stat Point estimator point estimator of a population parameter is a rule that tell us how to obtain a single number based on the sample data The resulting number is called a point estimator of this unknown population parameter A point estimator does not provide the information on the reliability of this estimator Sample variance is a point estimator of the population variance Sample standard deviation is a point estimator of the population standard deviation Unbiasedness mean of the sampling distribution of stats is called the expectation of this sample stats If the expectation of a sampling stats is equal to the population parameter this stats is intended to estimate the stats is an unbiased estimator of this population parameter If the expectation of a sampling statistics is not equal to the population parameter the statistics is a biased estimator Note 1 The sample mean is an unbiased estimator for the population mean 2 The sample median is a biased estimator for the population mean 3 If the population is normally distributed and the sample are randomly selected from this population the sample mean is the best unbiased estimator of the population mean 4 If the sampled population is extremely skewed and the sample size of the random sample is small the sample mean is not necessary be the best candidate which can be used to estimate the population mean 5 If the sample contains some extremely values sample mean need not be the best candidate for estimating the population mean Central Limiting Theorem If a random sample of size n is selected from a population with nite variance When the sample size is suf ciently large the sample mean will tend to be normally distributed Some properties of the sample mean The expectation of the sampling distribution of x is equal to the population mean u ie ux Ex u The standard error of the sample mean is ox Where the sample size is n and the population standard deviation is o The sampling distribution of sample mean is approximately normal for the suf ciently large sample even if the sampled population is not normally distributed The sampling distribution of mean is exactly normal if the sampled population is normally distributed no matter how small the sample size is

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