University:
California State University, Northridge
Course:
MSE 362 | Engineering Statistical Applications
Academic year:

2003
Views:

63

Pages:

28
Author:

kitkat23235o6

PROBABILITY DISTRIBUTIONS For basic probability concepts, had "simpie" random experiments with some specific QUtcomes and numerical values Now, define a Randoin Variable as a function that assigns a real number to each outcome in the sample space of a random experiment Further, define Discrete Random Variable and Continuous Random Variable Discrete Random Variable A random variable with a finite (or countably infinite) range (i.e., the set of possible values of a random variable X) Examples of discrete random variables: Number of scratches on a surface Number of transmitted bits received in error Proportion of defective parts among 1,000 tested Binomial Multinomial Hypergeometric Poission Geometric Pascal Negative Binomial Continuous Random Variable A random variable with an interval (either finite or infinite) of real numbers for its range Examples of continuous random variables: Length Pressure Height Time Distributions of continuous random variables include: Normal Gamma Exponential Beta Uniform Log-normal Rayleigh Cauchy Weibull Extreme value BINOMIAL DISTRIBUTION • Independent repeated trials (known as Bernoulli trials) • Each trial results in one of two possible outcomes {i.e., "success" or "failure") • Probability of "success," p, remains constant from one trial to the next (i.e., "sampling with replacement") EXAMPLE APPLICATIONS • Ten space shots are planned. Each has a probability of success of 0.95. What are the chances that at least nine shots will be successful? • Items are manufactured in large lots, from each of . which twenty units are selected at random. The lot is accepted if the sample contains three or fewer defective units. If the production process yields on the average 10 per cent defectives, what is the probability of lot acceptance? BINOMIAL DISTRIBUTION Binomial Probability Function: �.r O) I, .Z ) ••• , '11· > O�--"�.t. , Hu- �-4-tN ..,rJ �o..t.) X A.c) ��A/- AM,�-&4�.._,�- � )1- � )1,-buLW ��:rlu,��AU� ��.4.vMF�a-1., � .:ti-, ().,t:. X (>t) 1:' )t f):. = n.' "X. .' ( n .,. ;c ) .1 BINOMIAL DISTRIBUTION Distribution Function: The probability of x or fewer successes in n independent trials, given a probability of success p in a single trial: -8.., (x �):) = = r 6. ; f', X "n. > L. (;) 1° � (1--r)�-� ft:() F{r-)=O��t;, (1 J. :: ¥ (I ·-/J ) = )11"� '� j;r (1-+) BINOMIAL DISTRIBUTION _ Example Probability Functions: f(x) 0.4- f(x) 0.30.3 ,__ 0.2 ,_ 0.2 0.1._ 0.1 -O . I X O 1 2 3 4 5 6 7 8 9 10 n = IO, p = 0.75 0.2 0 I - f( x ) 0.1 � 0 - I ' 01 2 3 4 5 I . . I 6 7 8 9 10111213141516171819 X n = 40, p = 0.25 - ')1234 5 n = 5, p = 0.25 X BINOMIAL DISTRIBUTION Tabled Values: Table 1 Binomial Distribution Function 1' 0.05 0.10 0.15 0.20 0.25 0.30 0 0.9025 0.9975 0.8100 0.9900 0.7225 0.9775 0.6400 0.9600 0.5625 0.9375 0.4900 0.9100 3 0 0.8574 0.9927 0.9999 0.7 290 0.9720 0.9990 0.6141 0.9393 0.9966 0.5120 0.8960 0.9920 0.4219 0.8438 0.9844 0.3430 4 0 0.8145 0.9860 0.9995 0.6561 0.9477 0.9963 0.9999 0.5220 0.8905 0.9880 0.9995 0.4096 0.8192 0.9728 0.9984 0.3164 0.7383 0.9492 0.9961 0. 77 38 0.977 0.5905 n X .., - -' 'I 0 l 1.0000 0.9185 0.7840 0.9730 0. BINOMIAL DISTRIBUTION Example Problem: (�: ��-, ��) Samples of air each have a 10% chance of containing a particular molecule. Assume that the samples are independent with regard to the presence of the molecule. Find the probability that: ( a) Of 18 samples, exactly 2 contain the molecule, and (b) Of the 18 samples, at least 4 contain the molecule. � X =,>��CM/,,. � + o.,o� n,.: 1s c (tJ.) -/3., { X =- .Z) : ( �8) (o. I )2.(c. 9) 1 " 18 .' : �/ /,., (tJ�I) � (D.? • ) ,, :;; CJ . .291 � J9. � ¼ <�) �{X�Jf) = I- ffu(xsJ) • � ( X � .s) ::. C. 9018 = (), ...t/uu.; fJAJ (X � J/,).: I - �- 9tJl8 O 9 3 � ..-#"tJ "· 8¼ HYPERGEOMETRIC DISTRIBUTION • Probability of exactly x successes in n trials, when n is not small relative to the population size • Frequently applicable to problems involving sampling from a small lot (often referred to as "sampling without replacement") EXAMPLE APPLICATIONS • From an order for 25 high-reliability items, five are selected at random and life tested. If fewer than two items fail this test, the remaining 20 items will be accepted. Otherwise, the complete lot will be rejected. What is the probability of lot acceptance if four of 25 submitted items are defective? • A radio manufacturing company puts into a container 24 items for shipment. An inspector at the plant will test two items at random prior to shipping, and if any of the tested items prove defective he will return the whole container for examination. If there are actually two defectives in each container, what is the probability that a box will not be shipped? HYPERGEOMETRIC DISTRIBUTION Probability Function: Probability of exactly x defective units in randomly selecting n units from a lot of N units, of which exactly k are defective J(z) (� )(�=�) f r-x ; 1✓, 'JI, -Id = X : 0J /) (;!) .2 J • • •, � ; ): � � (>1-X) � {Al-�) Mean and Variance of Hypergeometric Distribution: �� =� = r,-A_/N 121·�.,\.�."a,n � = d'- i : n � (N - � )(Al - ?L ) N-1(N-I) POISSON DISTRIBUTION • Probability of exactly x independent occurrences in a given time period (or volume or area) • Assumes occurrences take place independently and at a constant rate EXAMPLE APPLICATIONS • In 10 random commuter train trips from Santa Barbara to Los Angeles, a total of 15 separate and independent delays were recorded. The train company claims that the long-run average number of delays per run is 1.2. If the number of delays is a Poisson variate, do the data refute the company's claim? • The average number of independent insulation breaks per yard in an enameling process is 0.07. What is the probability of finding one break in a piece of wire 16 yards long? l POISSON DISTRIBUTION Probability Function: �:: O.J l,.2 ) •.. ; �;? = >n,.<..a,r-J � Distribution Function: A> 0 -1- �A.L.-rlu.... F(:x..) = 0 � x 16,,,,,,.t""-) Contamination is a problem in the manufacture of optical storage disks. The number of particles of contamination that occur on an optical disk has a Poisson distribution, and the average number of particles per centimeter squared of media surface is 0.1. The area of a disk under study is 100 square centimeters. Find the probability that: (a) 12 particles occur in the area of this disk (b) no more than 12 particles occur in the area of this disk - - --- -· ��: -;,M-t:.Ca).c.� 1-t. ��'r h.,t,,,,J,' 4'0,lw., � (X�IJ.)-lai.(X.1:II) �(1'�1 )•= -/1.. (). 7 9 J. - 9 7 : �- tJ9S f). , GEOMETRIC DISTRIBUTION Probability Function: j(;r.; j,) r -fa x: I, � ;c: � "? � x (1--f r -l �J.3>· ••• ; O�� .1. Jtf::�,_..,) .rlu. � -t-lu- � � 'io.,uc� Example Probability Function: f(x) 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 Geon1etric distribution with p = -½ Mean and Variance of Geometric Distribution: #I GEOMETRIC DISTRIBUTION • Appropriate to problems in which we desire the probability that exactly x independent Bernoulli trials, each with a probability of "success," p, will be required until the first "success" is achieved • Special case of the Pascal distribution, which is a special case of the generalized negative binomial distribution EXAMPLE APPLICATION • The probability is 0.1 that a bit transmitted through a digital transmission channel is received in error. Assume the transmissions are independent events, and that the number of bits transmitted until the first error is a random variable. What is the probability that the first incorrect bit will be the fifth one? Table �I IJ rtri � ..' --c � 8 �---- � I (S�) (n)p•(I - p).-• •-o k p n 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0 0.9025 0.9975 0.8100 0.9900 0.7225 0.9775 0.6400 0.9600 0.5625 0.9375 0.4900 0.9100 0.4225 0.8775 0.3600 0.8400 0.3025 0.7975 0.2500 0.7500 0.2025 0.6975 0.1600 0.6400 0.1225 O.H75 . 0.0900 0.5100 0.0625 0.4375 0.0400 0.3600 0.0225 0.2775 0.0 l 00 0.1900 0.0025 0.0975 0 I 2 3 0.8145 0.9860 0.9995 1.0000 0.6561 0.9477 0.9963 0.9999 0.5220 0.8905 0.9880 0.9995 0.4096 0.8192 0.9728 0.9984 0.3164 0.7383 0.9492 0.9961 0.2401 0.6517 0.9163 0.9919 0.1785 0.5630 0.8735 0.9850 0.1296 0.4752 0.8208 0.9744 0.0915 0.3910 0.7585 0.9590 0.0625 0.3125 0.6875 0.9375 0.0410 0.2415 0.6090 0.9085 0.0256 0.1792 0.5248 0.8704 0.0150 0.1265 0.4370 0.8215 0.0081 0.0837 0.3483 0.7599 0.0039 0.0508 0.2617 0.6836 0.0016 0.0272 0.1808 0.5904 0.0005 0.0120 0.1095 0.4780 0.000 l 0.0037 0.0523 0.3439 0.0000 0.0005 0.0140 0.1855 X 0 4 �� fl. Binomial Distribution Function B(x; n, p) = .. �- r 1 6 0.8574 0.9927 0.9999 o 0.7738 I I 0.9774 2 0.9988 1.0000 4 1.0000 0 I 2 3 4 5 0 3 4 5 6 0.7351 0.9672 0.9978 0.9999 1.0000 1.0000 0.6983 0.9556 0.9962 0.9998 1.0000 1.0000 1.0000 o I o.6634 0.9428 0.9942 0.9996 4 1.0000 5 1.0000 6 1.0000 7 1.0000 0.7290 0.9720 0.9990 0.5905 0.9185 0.9914 0.9995 1.0000 0.5314 0.8857 0.9841 0.9987 0.9999 1.0000 0.4783 0.8503 0.9743 0.9973 0.9998 1.0000 1.0000 0.4305 0.8131 0.9619 0.9950 0.9996 1.0000 1.0000 1.0000 0.6141 0.9393 0.9966 0.4437 0.8352 0.9734 0.9978 0.9999 0.3771 0.7765 0.9527 0.9941 0.9996 1.0000 0.3206 0.7166 0.9262 0.9879 0.9988 0.9999 1.0000 0.2725 0.6572 0.8948 0.9786 0.9971 0.9998 1.0000 1.0000 0.5120 0.8960 0.9920 0.3277 0.7373 0.9421 0.9933 0.9997 0.2621 0.6554 0.9011 0.9830 0.9984 0.9999 0.2097 0.5767 0.8520 0.9667 0.9953 0.9996 1.0000 0.1678 0.5033 0.7969 0.9437 0.9896 0.9988 0.9999 1.0000 0.4219 0.8438 0.9844 0.2373 0.6328 0.8965 0.9844 0.9990 0.1780 0.5339 0.8306 0.9624 0.9954 0.9998 0.1335 0.4449 0.7564 0.9294 0.9871 0.9987 0.9999 0.1001 0.3671 0.6785 0.8862 0.9727 0.9958 0.9996 1.0000 0.3430 0.7840 0.9730 0.1681 0.5282 0.8369 0.9692 0.9976 0.1176 0.4202 0.7443 0.9295 0.9891 0.9993 0.0824 0.3294 0.6471 0.8740 0.9712 0.9962 0.9998 0.0576 0.2553 0.5518 0.8059 0.9420 0.9887 0.9987 0.9999 0.2746 0.7183 0.9571 ,0.2160 0.6480 0.9360 0.1160 0.4284 0.7648 0.9460 0.9947 0.0778 0.3370 0.6826 0.9130 0.9898 0.0754 0.3191 0.6471 0.8826 0.9777 0.9982 0.0490 0.2338 0.5323 0.8002 0.9444 0.9910 0.9994 0.0319 0.1691 0.4278 0.7064 0.8939 0.9747 0 9964 0.9998 0.0467 0.2333 0.5443 0.8208 0.9590 0.9959 0.0280 0.1586 0.4199 0.7102 0.9037 0.9812 0.9984 0.0168 0 1064 0.3154 0.594i 0.8263 0.9502 0.99 ! 5 0.9993 0.1664 0.5748 0.9089 0.0503 0.2562 0.5931 0.8688 0.9815 0.0277 0.1636 0.4415 0.7447 0.9308 0.9917 0.0152 0.1024 0.3164 0.6083 0.8471 0.9643 0.9963 0.0084 0.0632 0.2201 0.4770 0.7396 0.9115 0.98!9 0.9983 0.1250 0.5000 0.8750 0.0313 0.1875 0.5000 0.8125 0.9688 0.0156 0.1094 0.3438 0.6563 0.8906 0.9844 0.0078 0.0625 0.2266 0.5000 0.7734 0.9375 0.9922 0.0039 0.0352 0.1445 0.3633 0.6367 0.8555 0.9648 0.9961 0.0911 0.4252 0.8336 0.0185 0.1312 0.4069 0.7438 0.9497 0.0083 0.0692 0.2553 0.5585 0.8364 0.9723 0.0037 0.0357 0.1529 0.3917 0.6836 0.8976 0.9848 0.0017 0.0181 0.0885 0.2604 0.5230 0.7799 0.9368 0.9916 0.0640 0.3520 0.7840 0.0102 0.0870 0.3174 0.6630 0.9222 0.0041 0.0410 0.1792 0.4551 0.7667 0.9533 0.0016 0.0188 0.0963 0.2898 0.5801 0.8414 0.9720 0.0007 0.0085 0.0498 O.i737 0.4059 0.6846 0.8936 0.9832 0.0429 0.2818 0.7254 0.0053 0.0540 0.2352 0.5716 0.8840 0.0018 0.0223 O."ll74 0.3529 0.6809 0.9246 0.0006 0.0090 0.0556 0.1998 0.4677 0.7662 0.9510 0.0002 0.0036 0.0253 0.1061 0.2936 0.5722 0.8309 0.9681 0.0270 0.2160 0.6570 0.0156 0.1563 0.5781 0.0024 0.0308 0.1631 0.4718 0.8319 0.0010 0.0156 0.1035 0.3672 0.7627 0.0007 0.0109 0.0705 0.2557 0.5798 0.8824 0.0002 0.0038 0.0288 0.1260 0.3529 0.6706 0.9176 0.0001 0.0013 0.0113 0.0580 0.1941 0.4482 0.7447 0.9424 0.0002 0.0046 0.0376 0.1694 0.4661 0.8220 0.0001 0.0013 0.0129 0.0706 0.2436 0.5551 0.8665 0.0000 0.0004 0.0042 0.0273 0.1138 0.3215 0.6329 0.8999 0.0080 0.1040 0.4880 0.0003 0.0067 0.0579 0.2627 0.6723 0.0001 0.0016 0.0170 0.0989 0.3446 0.7379 0.0000 0.0004 0.0047 0.0333 0.1480 0.4233 0.7903 0.0000 0.0001 0.0012 0.0104 0.0563 0.2031 0.4967 0.8322 0.0034 0.0607 0.3859 0.0001 0.0022 0.0266 0.1648 0.5563 0.0000 0.0004 0.0059 0.0473 0.2235 0.6229 0.0000 0.0001 0.00 l 2 0.0121 0.0738 0.2834 0.6794 0.0000 0.0000 0.0002 0.0029 0.0214 0. I 052 0.3428 0.7275 0.0010 0.0280 0.2710 0.0000 0.0005 0.0086 0.0815 0.4095 0.0000 0.0001 0.0013 0.0158 0.1143 0.4686 0.0000 0.0000 0.0002 0.0027 0.0257 0.1497 0.5217 0.0000 0.0000 0.0000 0.0004 0.0050 0.0381 0. i 869 0.5695 0.0001 0.0073 0.1426 0.0000 0.0000 0.0012 0.0226 0.2262 0.0000 0.0000 0.0001 0.0022 0.0328 0.2649 0.0000 0.0000 0.0000 0.0002 0.0038 0.0444 0.3017 0.0000 0.0000 0.0000 0.0000 0.0004 0.0058 0.0572 0.3366 Table 2 Poisson Distribution Function• F(x; }.) 0 � I 1.000 2 0.02 0.04 0.06 0.08 0.10 0.980 0.961 0.942 0.923 0.905 0.15 0.20 0.25 0.30 0.861 0.990 0.999 0.819 0.982 0.999 0.35 0.40 0.45 0.50 0.999 0.998 0.997 0.995 0.974 3 = • L e- ••o 4 1 S"-;,a.�) ( )! -k, 5 1 6 1.000 1.000 1.000 1.000 0.998 0.996 0.705 0.670 0.638 0.607 0.951 0.938 0.925 0.910 0.994 0.992 0.989 0.986 1.000 0.999 0.999 0.998 1.000 1.000 1.000 0.55 0.60 0.65 0.70 0.75 0.577 0.549 0.522 0.497 0.472 0.894 0.878 0.861 0.844 0.827 0.982 0.977 0.972 0.966 0.959 0.998 0.997 0.996 0.994 0.993 1.000 1.000 0.999 0.999 0.999 1.000 1.000 1.000 0.80 0.85 0.90 0.95 1.00 0.449 0.427 0.407 0.387 0.368 0.809 0.791 0.772 0.754 0.736 0.953 0.945 0.937 0.929 0.920 0.991 0.989 0.987 0.984 0.981 0.999 0.998 0.998 0.997 0.996 1.000 1.000 1.000 1.000 0.999 1.000 I.I 1.2 1.5 0.333 0.301 0.273 0.247 0.223 0.699 0.663 0.627 0.592 0.558 0.900 0.879 0.857 0.833 0.809 0.974 0.966 0.957 0.946 0.934 0.995 0.992 0.989 0.986 0.981 0.999 0.998 0.998 0.997 0.996 1.000 1.000 1.000 0.999 0.999 1.000 1.000 1.6 1.7 1.8 1.9 2.0 0.202 0.183 0.165 0.150 0.135 0.525 0.493 0.463 0.434 0.406 0.783 0.757 0.731 0.704 0.677 0.921 0.907 0.891 0.875 0.857 0.976 0.970 0.964 0.956 0.947 0.994 0.992 0.990 0.987 0.983 0.999 0.998 0.997 0.997 0.995 1.000 1.000 0.999 0.999 0.999 1.3 1.4 9 1.000 1.000 1.000 0.963 0.779 0.741 8 1.000 1.000 1.000 1.000 • Reprinted by ki11d permissio11Jrom E. C. Molina, Poisson's Exponential Binomial Limit, D. Van Nostrand Company. Inc., Princeton, N.J., 1947. Jau.M:.L-: �'. {e:,�tl.J?.oi. a. 'rr� -A,,n,d ..J-/'I..Uvr,� f'�,a,-,,i,olM,�c.s flh.- &,�. '!°��-Ea.LL > ;q9�

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