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by: Helga Torp Sr.


Marketplace > University of Kentucky > Statistics > STA 320 > INTRODUCTORY PROBABILITY
Helga Torp Sr.
GPA 3.87


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This 7 page Class Notes was uploaded by Helga Torp Sr. on Friday October 23, 2015. The Class Notes belongs to STA 320 at University of Kentucky taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/228279/sta-320-university-of-kentucky in Statistics at University of Kentucky.




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Date Created: 10/23/15
Continuous Distributions STA320 Summer ll 2004 1 Introduction The motivation for continuous random variables is that we want the sample space to be every number in some interval of the real line For example if we are measuring a person s height we consider any real number between 0 and 12 feet to be a possible value The dif culty is that we CANNOT place a nonzero probability on every real number in an interval There are uncountably many numbers in any interval on the real line For mathematical reasons the sum of uncountably many positive numbers is in nity This is in contrast to a countably in nite number of positive numbers which can have a nite sum In particular that sum can be 1 so that PS 1 Therefore when we want the sample space to be an interval we have to consider a different way of assigning probabilities than placing positive probability on individual points Furthermore there is some intuitive justi cation that no point should have positive probability Some proportion of people are between 59 and 61 feet The proportion of people who are exactly 6 feet must be smaller since being exactly 6 feet tall is a subset of being between 59 and 61 feet When we replace the interval from 59 to 61 feet with 599 to 601 feet or 5999 to 6001 feet we gradually acquire smaller and smaller subsets of people In the limit this leaves no probability left for a person to be exactly 6 feet This argument may be formalized by looking at the cdf of a continuous random variable DEFINITION Continuous Random Variable A random uariable X is continuous if its cdf is a continuous function There are random variables that are neither discrete nor continuous Take R to be the amount of rainfall on a given day There is a nonzero probability of getting 0 rainfall on a given day so the cdf of R has a jump at 0 resulting in a discontinuous cdf However if there is rain on a particular day it could be any amount between 0 and in nity Surely there is some theoretically maximum rainfall for a day but I don t know what it is so I ll use in nity The point is that the rainfall total B can be anywhere in an interval from 0 to that theoretical maximum If the sample space includes an interval then the random variable cannot be discrete The result is that R is neither discrete nor continuous In this course we will almost exclusively focus on discrete and continuous random variables ignoring random variables such as R THEOREM IfX is a continuous random uariable then for any real number x PX x 0 To prove this recall by de nition a continuous random variable has a continuous cdf Let x E R be xed and for each 6 gt 0 let A6 be the event X E x 7 6 Notice X x Q A6 for all e and thus PX x PA5 for all 6 Using the theorem involving computing interval probabilities with CDFs PA5 Px7 elt X g x Fxx7Fxxie Since FX is a continuous function 1 P A 0 51g a This implies that for any p gt 0 there exists an A6 such that PA5 lt p Furthermore we already established PX z lt PA5 for all 6 Thus for all p gt O PX z lt p Since all probabilities are nonnegative this implies PX z 0 This result has some very untuitive consequences In particular we have just said that every number in the sample space has 0 probability of occuring So how can anything happen Intervals still have probability of occuring If a lt b then it is still possible that Pa lt X g b FXb 7 FXa will be greater than 0 Thus intervals have probability but individual points do not One fortunate aspect of continuous random variables is that all interval probabilities have the same form THEOREM IfX is a continuous random variable then FXb7FXaPaltX bPaltXltbPa XltbPa Xltb 1 To prove this result recall the theorem that stated how to compute interval probabilities in general using a cdf All the interval probabilities involved FXb 7 FXa and then we either added PX a or subtracted PX b depending on whether a or b was included in the interval When X is a continuous random variable both PX a 0 and PX b 0 so we do not need to worry about including them in the equation To facilitate calculating interval probabilities we use a probability density function or pdf DEFINITION Probability density function or pdf Let X be a continuous random variable with cdf A function fX that satis es FXz mt dt for all real numbers m is called a probability density function or pdf of X THEOREM If is a pdf of X then PaltX bbezdx The proof follows from the de nition of pdf 17 a b b Pa lt X g b FXb 7 FXa mt dt 7 mt dt mt dt fxz dz 700 700 a a If we have the cdf of X then it is typically easier to compute interval probabilities using Unfortunately for most common continuous distributions all the families discussed in this handout for example there is an analytical form for the pdf fXm but not the cdf Thus we are forced to describe probabilities in terms of the pdf fXm and often the exact probabilities must be computed by numerical integration EXAMPLE Let X be a random variable with cdf 0 x 0 zZ4 0 lt m lt 2 2 1 22 A pdf ofX is xQ 0 lt m lt 2 0 otherwise fxm 9021o2 Recall an indicator function Az is equal to 1 if z E A and 0 otherwise To show that the g given is a pdf we have to show that E mt dt Fm 00 Since is split into pieces we have check the equation in pieces For x g 0 nowwmmmowo The last equality follows because the indicator function zeroes out77 the pdf for everywhere except between 0 and 2 Since the interval foo z does not include this region we assumed x g 0 the integrand is 0 everywhere and thus the entire integral is O which agrees with the cdf For 0 lt m lt 2 E E E new Mansowamwww4 700 700 0 Again the indicator function zeroes out the entire integrand except in the region from 0 to 2 and the integral of the pdf equals the cdf Finally for z 2 2 lnowwmmsow mwl Because the equation in the de nition of the pdf is veri ed we have established that fXm is the pdf of X THEOREM If 239s di erentz39able except at a countable set of points 1 2 then the function at fX FXW where 239s di erentz39able and 0 where is not di erentz39able is a pdf of X This theorem follows directly from the Fundamental Theorem of Calculus For a differentiable F m d a FXt dt Fm 7 FXa 1 3 By de nition 1 700 a Fxt dt 7 11111100 Fxa where the last equality follows from noting the cdf tends to 0 as x tends to negative in nity The statement FXz is differentiable except at a countable set of points m1m2 is a technical point allowing a few discontinuities such as those in equation 2 in the example The cdf is not differentiable at z 0 and at z 2 but is differentiable everywhere else THEOREM If is a pdf of X then 1 there is no interval a b where lt 0 for m E a b 2 H mm dz 1 These results follow from the axioms of probability Recall that interval probabilities can be found by integrating the pdf If the pdf were negative over an interval we could integrate over that interval and acquire a negative probability which is impossible by axiom Furthermore axiom 2 states the probability of the sample space is equal to 1 For a random variables the sample space consists of numbers and thus the interval from negative in nity to in nity contains the entire sample space and thus integrates to 1 2 Common Continous Distributions Like discrete distributions there are a number of commonly used families of continuous distribu tions Here we will only discuss three There are many other commonly used families which we omit here such as the t and F families 21 The Gamma Function The integral 00 mail exp7z dz 0 for 04 gt 0 cannot be solved analytically Nevertheless this integral appears often in the sciences mathematics and statistics In statistics the integral appears in probability densities Since there is no analytical form of the integral we often just de ne its value to be Pa Thus by de nition 00 Ha mail expiz dz 3 0 Note that since the integrand ma l exp7z is greater than zero for all z gt 0 Pa gt 0 for 04 gt 0 there are mathematical de nitions of Pa for 04 g 0 but we do not consider that here We can actually form a more complicated integral using integration by substitution Let y zB Then z By and dy 13 dm resulting in ma KW epoymdy 0 Barf exp ydy and thus A00 if exp y dy 1 4 The Gamma function has three properties you should know H P1 1 By de nition 00 CO P1 zl 1 exp7z dz exp7z dz 7exp7zlg0 exp0 1 0 0 2 For 04 gt 1 Na 1 04 Na This can be seen using integration by parts 00 Na 1 z exp7z dz 0 Let z and dv exp7z dz Then du ozzuquot1 dz and vz 7 exp7z Thus 000 z exp7z dz 000 u dv 7 00 Uz du 0 704 z quot1 exp7z 0 04 z quot1 exp7z dz 04 z quot1 exp7z dz 04 Na 0 0 3 For an integer Oz 2 0 Na 1 a Proceed recursively with PltOz 1 04POz aa 71Pa 71 aa 71a 7 2POz 7 2 aa71a721P1al Note this last item establishes the P function as a continuous approximation to factorials The reason it is important that 04 be an integer above is that you need the recursion to end at P1 which we know to be 1 If we were evaluating F15 in contrast that would just be 05P05 and we wouldn t know the base case77 actually P05 but in general we don t know the base case 22 Gamma Distributions Equation 4 has a function that integrates to 1 and is nonnegative everywhere Thus the integrand in 4 forms a density that we call the demda 3 distribution where BBL m a71 exp 5 I0oo fXWlm The Gamma distribution and some variants ofthe Gamma distribution are used in reliability theory which studies how long appliances automobiles etc last until they fail The Gamma distribution is de ned on 0 oo conveniently where time until failure77 is de ned For most values of 04 the cdf of X does not have an analytical form That is there is no common function whose antiderivative is the Gamma density However for 04 1 we arrive at a special case of the Gamma distribution known as the Exponential distribution written X N Ezp which has pdf fXMB exp 10oo The relationship between the Gamma and Exponential distributions is analogous to the relationship between the Negative Binomial and Geometric distributions The Exponential distribution is a special case of the Gamma distribution If X N Ezp then 1 0 exp7 tdt iexp7 tlg iexp7 z 1 17 exp7 z Note by the complement rule that PX gt z 17FXm exp7 z The exponential distribution has a particularly nice property in that it is memoryless meaning 7PXgtts XgttiPXgtts PXgttlegtti PXgtt 7 PXgtt amwam emem A part or appliance or whatever else whose failure time follows a Exponential distribution thus has the memoryless property It doesn t matter how old the part is the probability it will fail in the next 3 time units is always the same Thus the machine does not show wear in any sense Although somewhat unintuitive the exponential distribution does model the failure times for many electronic components exp7 s PX gt s 23 Beta Distributions A Betaa 3 distribution has density Na 5 71 1971 fX95 W 950 1 9 0195 Just like every discrete family of distributions provides a summation formula for free every con tinuous distribution provides an integration formula for free Here since dz 1 we nd 135071 7351971 35 NONFB A 1 l d mmmm The Beta distribution places probability on the interval 01 and is thus a very common choice for modelling probabilities which must also be between 0 and 1 A special case of the Beta distribution is achieved by setting 04 1 and B 1 Replacing these in the density we get a standard uniform distribution which has density fX 11o1 24 Normal Distributions A Normal distribution has two parameters In and 02 The parameter In is called the mean and determines the location of the distribution The parameter 02 is called the variance 0 the square root of 02 is the standard deviation and determines the spread of the distribution If a random variable X has a normal distribution with mean u and variance 02 then we write X N Nu 02 One particular normal called the standard normal occurs when u 0 and 02 1 Usually a random variable with a standard normal distribution is written as Z The density function fXm for a normal distribution is rather messy The range of a normal distribution is the entire real line foo 00 and the density is 1 7 x7 2 fXWMTZ WBXP Plotting this function results in a standard bell curve77 shape The parameter 2 controls the highest point of the density and the density fXm is symmetric around In The parameter 0 controls the spread of the distribution The in ection points of the normal density occur at u i 0 Unfortunately like the Gamma and Beta distributions it is not possible to analytically deter mine the cdf of a normal distribution and thus there is no analytic formula for nding normal probabilities In particular the quantity 7 PaltXltb my 1 cannot be solved analytically Thus one must use numerical integration such as the trapezoidal rule or Simpson s rule from your calculus classes At some point people did this for the standard normal which has density The numerical integration was focused at nding the cumulative distribution function of the stan dard normal which is Recall from the previous sections that knowing the cumulative density functions allows you to compute all kinds of interval probabilities such as PZ 2 PZ gt z and Pa lt Z b The numerical results for ltIgtz were typically reported in tables Today we of course commonly use computer programs to nd these probabilities a topic we will not cover in this course Normal tables provide values of ltIgtz for 2 between about 35 and 35 For values of z lt 735 PZ z is essentially O and for values of z gt 35 PZ z is essentially 1 If you are working problems where z is outside the range of the table you may substitute 0 or 1 appropriately for lf 2 is not divisible by 001 just use the closest 2 value in the table For example if you are trying to nd PZ 23125 just use PZ 231 Computer programs by contrast allow you to put in any number and get an answer and are thus easier to use and more accurate surprise


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