University:
California State University, Northridge
Course:
MATH 440A | Mathematical Statistics I
Academic year:

2003
Views:

427

Pages:

20
Author:

anilhetzlrf

PROBABILITY DENSITIES CONTINUOUS DISTRIBUTIONS NORMAL DISTRIBUTION @ NORMAIJ (GAUSSIAN) DISTRIBUTION Introduced by C. F. Gauss in connection with the theory of errors of physical measurements The most important continuous distribution, for the following reasons: • Many random variables that appear in connection with practical experiments or observations are normally distributed, and many other types of variables are approximately normally distributed • Even if a variable is neither normally nor approximately normally distributed, it can frequently be transformed into a normally distributed variable by a relatively simple transformation • Certain more complicated distributions can be approximated by the normal distribution (e.g., the binomial distribution). • Several variables that are used in statistical inference are normally or approximately normally distributed� NORMAL (GAUSSIAN) DISTRIBUTION EXAMPLE APPLICATIONS • Given that the height of a population is a normal random variable with mean 68 inches and standard deviation 2.5 inches. If we select 200 of these persons at random, how many of them can we expect to be (a) 68 inches, (b) greater than 70 inches, assuming that the measurements are recorded to the nearest inch? • Suppose the current measurements in a strip of wire are assumed to follow a normal distribution with a mean of 10 milliamperes and a variance of 4 milliamperes squared. What is the probability that a measurement will exceed 1.3 milliamperes? • The fill volume of an automated filling machine used for filling cans of carbonated beverage is normally distributed with a mean of 12.4 fluid ounces and a standard deviation of 0.1 fluid ounce. Determine the specifications that are symmetric about the mean that include 99% of all cans. • The sick-leave time of employees in a firm in a month is normally distributed with a mean of 100 hours and a standard deviation of 20 hours. How much time should be budgeted for sick leave if the budgeted amount should be exceeded with a probability of only 10%? {!f) NORMAL (GAUSSIAN) DISTRIBUTION J(',r.J:µ i,. a. • ti �� Probability Density Function: Distribution Function: Mean and Variance: �=r �a.,,,.-,�,: 0-J.., -(7( ?' ) .ava .._ ® NORMAL (GAUSSIAN) DISTRIBUTION EXAMPLE PR.OBABILITY DENSITY FUNCTIONS Normal distributions with different values ofµ and same u. p = be�J 0.7 ,./JII� 0-= ....,,. f!'116&----� 1'=0, v=0.5 0.6 0.5 IU 2.0 3.0 Normal distributions with different values of a and same µ. NORMAL ((;AUSSIAN) DISTRIBUTION DETERMINING PROBABILITIES: Bv(a< X �--t-) J = F(-t) - F(tL) -' 1 = (rr{:r/T' r ex.p --} (:1-;:-/ "- 1-1 � a. STANDARD NORMAL VARIATE: If x is normally distributed with arbitrary mean and variance, then the standardized or standard normal variate z = (x:-µ) I a is normally distributed withµ = 0 and a = 1. This allows us to use a single table to evaluate cumulative probabilities for any normal distribution. (j) NORMAI_J (GAUSSIAN) DISTRIBUTION TABLED VALUES: (.J�,. 'n,��tJ te..,,,,r') ) e) greater than 2.16 : o'r'\�(X � I. So) d) greater than -1.75 .1l r.,- -f -1.� 0 -f'A (X >!LI�')= O.OIS'f( ') J - ,Pl\- ( )( '- :l • I (. ) l�) .p.,__ (K � - �- f <.) .Pn.( x > -1.1s; � o.qsq<) (,) , - -f',-_,(x �f- 1.1s) ( 2.) fl\..(.�� (.1S) NORMAL (GAUSSIAN) DISTRIBUTION EXAMPLE P.ROBLEMS - - - If a random variable has the standard normal distribution, find the probability that it will take on a value a) between O and 2. 7 b) between 1.22 and 2.43 c) between -1.35 and -0.35 d) betwE�en -1.70 and 1.35 -f'�(o�x,<�.1): o.4-91-S -f,-. (�t.. ( - I. 3 S -J,1o o +-1,35 < X < - o. 3S): o. 11 '+'1 NORMAL (GAUSSIAN) DISTRIBUTION EXAMPLE l'ROBLEMS - - The time required to assemble a piece of machinery is a random variable having approximately a normal distribution with µ = 12.9 minutes and a = 2.0 minutes. What are the probabilities that the assembly of a piece of machinery of this kind will take a) at least 11.5 minutes? b) anywhere from 11.0 to 14.8 minutes? iµ X � �W#f? i-t;. � � u."1.,_ � '1Arf �' �J =la,9 �� ; 1?_ � (1 ,1. 9, JI.. tJ) 11) �(Xr4tha,,J.//.S�) IVQN...�&) =- f'� { X )} I/. S ) z = _'X.-fa_: tr //.S-IJ,9 =- - 0,7 ,J,O = �. � (X� //.S) 7S'J> fh. ( � ):,. - 0 . 7 ) -= 0 • i S-i - C ,1 O -t) -&(11.o/X1/'f.l) 2:.,t :: ( 11.0-1.2,.9 )/.2.0: - .2 A, o. 95" = (1'1'.3-/�-9)/,2.-�, -,..().95 -o.'lS' 0 t-o,qs � ¥,C NORMAL (GAUSSIAN) DISTRIBUTION EXAMPLE P�ROBLEMS - - Specifications for a certain job call for washers with an inside diameter of 0.300 ± 0.005 inch. If the inside diameters of the washers supplied by a given manufacturer may be looked upon as a random variable having the normal distribution with µ = 0.302 inch and a = 0.003 inch, what percentage of these washers will meet specifications? 'u.� � :.� �- .Jo� .;. c. o�s : o. .JDS' �� :: o. .Joe - (). tJtJS = O. �'IS 2µ = (IJ. J(JS - tJ. 3o�) / (). �o 3 :: /. oo t.,t_ = < I). .2. 95 - o. .3.bJ. >/ o. ()tJ.3 = - ,2. s8 � (D. J9S -� -x. � IJ,4os) = o.K.Jli -�.33 0 1.00 .. 6) NORMAL (GAUSSIAN) DISTRIBUTION EXAMPLE PROBLEMS: ($�"'•' �"'w�) A food processor packages instant coffee in small jars. The weights of the jars are normally distributed with a standard deviation of 0.3 ounce. If 5% of the jars weigh more than 12.492 ounces, 'What is the mean weight of the jars? � X � 3-Mi �� J-JnJ.J1.u_ ��) 1 = /:}. 11-'I :J -:µ '& (X >Jti. �9-2) = o. oS- � � (X .1 I.?. {/-9.2,)= I- tJ.tJS J/Urrn.J � cJ'�,) i:: o, 9S o. 9S ��e,(A1,}/4 � = /. & J/S � /. t, '-1-S= /�. '-1-9:J. -µ 0.'3 ��=II. 99 'I� I� .HWnc.u-) @ NORMAIJ (GAUSSIAN) DISTRIBUTION TRANSFORMATION OF DATA Although obs•�rvation values may not be Normally distributed, it is possible that some transformation of the data set might be Normally distributed If the transformed data can be demonstrated to be approximately Normally distributed, then the transformed data set can be evaluated and inferences drawn Typical transformations attempted include: -% -,

Continous Distributions

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