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# Introduction to Mathematical Statistics I ST 421

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This 14 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 421 at North Carolina State University taught by Thomas Gerig in Fall. Since its upload, it has received 15 views. For similar materials see /class/223953/st-421-north-carolina-state-university in Statistics at North Carolina State University.

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Date Created: 10/15/15

Continuous Random Variablesl Introduction 41 Probability distribution 42 Expected value 43 Uniform distribution 44 Normal distribution 45 Gamma distribution 46 Beta distribution 47 Moments 49 Tchebysheff s theorem 410 Introduction I A continuous rv takes an uncountably in nite numbers of values The probability distribution for a discrete rv can always be given by assigning a positive probability to each of the possible values of the variable and the sum of probability is 1 Unfortunately the probability distribution for a continuous rv cannot be speci ed in the same way It is mathematically impossible to assign nonzero probabilities to all the points on a line interval and the same time satisfy the requirement that the probabilities of the distinct possible values sum to 1 Probability distribution I De nition Let Y denote any random variable the distribution function of Y denoted by F g is given by for oo lt y lt 00 Theorem Propeties of a Distribution Function If F is a distribution function then 1 F oo 2 limynOOFy 2 0 2 Foo 2 limynOOFQ 2 1 3 F is a nondecreasing function of y If 31 and 32 are any values such that 31 lt 32 then PM E Pia2 De nition Let Y denote a rv With distribution function F Y is said to be continuous if the distribution function F is continuous for oo lt y lt 00 De nition Let F Y be the distribution function for a continous rv Y Then given by y 51 my Wherever the derivative exists is called the probability density function for the rv Y Theorem Properties of a Density Function If f is a density function then 1 y 2 0 for any values of y 2 If fydy 1 Theorem If the random variable Y has density function y and a g b then the probability that Y falls in the interval 0 b is b Pmsyswffm Approximating the Binomial with the Poisson Suppose Y is a Binomial with parameters n total number of trials and 19 probability of success For large n and small 1 such that A 2 up is less than 7 the following approximation can be used PY 2 7 N e for A 2 up The Poisson is the probability distribution of the number Y of rare events that occur in a xed unit of m or any other dimensions Where A is the average value of Y ie the rate number of occurrences per that unit of m Where the two units of time are the same Suppose that Y is the number Y of accidents in a 3 hours Window and A is the rate number of accidents per hour Then the probability that Y is 7 is Y is a Poisson With parameter 3 Moment generating functionl The parameters a and 0 are meaningful numerical descriptive measures that locate the center and describe the spread associated with a rv Y However they do not provide a unique characterization of the distribution of Y Many different distributions possess the same means and standard deviations We now consider a set of numerical descriptive measures that uniquely determine De nition The 2th moment of a random variable Y taken about the origin is de ned to be Em and denoted by a De nition The ith moment of a random variable Y taken about its mean or the ith central moment of Y is de ned to be EKY W and is denoted by ai De nition The moment generating function mt for a random variable Y is de ned to be EetY We say that a moment generating function for Y exists if there exists a positive constant 6 such that Mt is nite for It 3 b De nition If mt exists then for any positive integer k dkmt Wlto 2 WWW 2 life In other words if you nd the kth derivative of mt With respect to t and then set t 0 the result will be MC Method of Transformationsl Univariate case If we are given the density function of a rv Y7 the method of transformations results in a general expression for the density of U 2 MY for an increasing or decreasing function Bivariate case If Y1 and Y2 have a bivariate distribution and assume U 2 MY Then we nd the joint density of Y1 and U By integrating over 31 we nd the probability density function of U Let Y have probability density function If is either increasing or decreasing for all 3 such that fyy gt 0 then U 2 MY has density function dh l du fUW 2 fr h1u Where dh l dh1u du du Direct application of this method requires that the function be either increasing or decreasing for all 3 such that fyy gt 0 Steps for the Transformation method Let U 2 MY Where is either increasing or decreasing for all 3 such that fY3 gt 07 1 Find the inverse function y h1u 2 Evaluate dh l 0301 1 du du 3 Find by

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