Probability Examples of Discrete Probability Distribution

8/13/2019 Probability Examples of Discrete Probability Distribution

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No of Head in 6 Flips of Fair coin Probability Probability in % E(X)0 0.015625 1.56% 01 0.09375 9.38% 0.093752 0.234375 23.44% 0.468753 0.3125 31.25% 0.93754 0.234375 23.44% 0.9375

5 0.09375 9.38% 0.468756 0.015625 1.56% 0.09375

3

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

0 1 2 3 4 5 6

Discrete Probability Distribution

Probability in %


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No of Head Probability Probability in %0 0.000000000000000000000000000001 0.0000000000000000000000000000011 0.000000000000000000000000000079 0.0000000000000000000000000000792 0.000000000000000000000000003905 0.0000000000000000000000000039053 0.000000000000000000000000127559 0.0000000000000000000000001275594 0.000000000000000000000003093301 0.000000000000000000000003093301

5 0.000000000000000000000059391381 0.0000000000000000000000593913816 0.000000000000000000000940363535 0.0000000000000000000009403635357 0.000000000000000000012627738903 0.0000000000000000000126277389038 0.000000000000000000146797464748 0.0000000000000000001467974647489 0.000000000000000001500596306315 0.000000000000000001500596306315

10 0.000000000000000013655426387463 0.00000000000000001365542638746311 0.000000000000000111726215897425 0.00000000000000011172621589742512 0.000000000000000828636101239238 0.00000000000000082863610123923813 0.000000000000005609228993004070 0.00000000000000560922899300407014 0.000000000000034857351599382500 0.00000000000003485735159938250015 0.000000000000199848815836459000 0.000000000000199848815836459000

16 0.000000000001061696834131190000 0.00000000000106169683413119000017 0.000000000005246031415707060000 0.00000000000524603141570706000018 0.000000000024190033750204800000 0.00000000002419003375020480000019 0.000000000104399093027200000000 0.00000000010439909302720000000020 0.000000000422816326760158000000 0.00000000042281632676015800000021 0.000000001610728863848220000000 0.00000000161072886384822000000022 0.000000005783980920182250000000 0.00000000578398092018225000000023 0.000000019615239642357200000000 0.00000001961523964235720000000024 0.000000062932227185896000000000 0.00000006293222718589600000000025 0.000000191313970645124000000000 0.00000019131397064512400000000026 0.000000551867223014780000000000 0.00000055186722301478000000000027 0.000001512524981596060000000000 0.00000151252498159606000000000028 0.000003943368702018310000000000 0.00000394336870201831000000000029 0.000009790432639493740000000000 0.00000979043263949374000000000030 0.000023170690580135200000000000 0.00002317069058013520000000000031 0.000052320914213208400000000000 0.00005232091421320840000000000032 0.000112816971272231000000000000 0.00011281697127223100000000000033 0.000232471334742779000000000000 0.00023247133474277900000000000034 0.000458105277287240000000000000 0.00045810527728724000000000000035 0.000863855665741652000000000000 0.00086385566574165200000000000036 0.001559739396477980000000000000 0.00155973939647798000000000000037 0.002697927604718680000000000000 0.00269792760471868000000000000038 0.004472879976244120000000000000 0.004472879976244120000000000000

39 0.007110732269926550000000000000 0.00711073226992655000000000000040 0.010843866711638000000000000000 0.01084386671163800000000000000041 0.015869073236543400000000000000 0.01586907323654340000000000000042 0.022292269546572900000000000000 0.02229226954657290000000000000043 0.030068642644214600000000000000 0.03006864264421460000000000000044 0.038952559789096100000000000000 0.03895255978909610000000000000045 0.048474296626430800000000000000 0.04847429662643080000000000000046 0.057958398140297600000000000000 0.05795839814029760000000000000047 0.066590499990980200000000000000 0.066590499990980200000000000000


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48 0.073527010406707400000000000000 0.07352701040670740000000000000049 0.078028664105077200000000000000 0.07802866410507720000000000000050 0.079589237387178700000000000000 0.07958923738717870000000000000051 0.078028664105077200000000000000 0.07802866410507720000000000000052 0.073527010406707400000000000000 0.07352701040670740000000000000053 0.066590499990980200000000000000 0.06659049999098020000000000000054 0.057958398140297600000000000000 0.057958398140297600000000000000

55 0.048474296626430800000000000000 0.04847429662643080000000000000056 0.038952559789096100000000000000 0.03895255978909610000000000000057 0.030068642644214600000000000000 0.03006864264421460000000000000058 0.022292269546572900000000000000 0.02229226954657290000000000000059 0.015869073236543400000000000000 0.01586907323654340000000000000060 0.010843866711638000000000000000 0.01084386671163800000000000000061 0.007110732269926550000000000000 0.00711073226992655000000000000062 0.004472879976244120000000000000 0.00447287997624412000000000000063 0.002697927604718680000000000000 0.00269792760471868000000000000064 0.001559739396477980000000000000 0.00155973939647798000000000000065 0.000863855665741652000000000000 0.000863855665741652000000000000

66 0.000458105277287240000000000000 0.00045810527728724000000000000067 0.000232471334742779000000000000 0.00023247133474277900000000000068 0.000112816971272231000000000000 0.00011281697127223100000000000069 0.000052320914213208400000000000 0.00005232091421320840000000000070 0.000023170690580135200000000000 0.00002317069058013520000000000071 0.000009790432639493740000000000 0.00000979043263949374000000000072 0.000003943368702018310000000000 0.00000394336870201831000000000073 0.000001512524981596060000000000 0.00000151252498159606000000000074 0.000000551867223014780000000000 0.00000055186722301478000000000075 0.000000191313970645124000000000 0.00000019131397064512400000000076 0.000000062932227185896000000000 0.00000006293222718589600000000077 0.000000019615239642357200000000 0.00000001961523964235720000000078 0.000000005783980920182250000000 0.00000000578398092018225000000079 0.000000001610728863848220000000 0.00000000161072886384822000000080 0.000000000422816326760158000000 0.00000000042281632676015800000081 0.000000000104399093027200000000 0.00000000010439909302720000000082 0.000000000024190033750204800000 0.00000000002419003375020480000083 0.000000000005246031415707060000 0.00000000000524603141570706000084 0.000000000001061696834131190000 0.00000000000106169683413119000085 0.000000000000199848815836459000 0.00000000000019984881583645900086 0.000000000000034857351599382500 0.00000000000003485735159938250087 0.000000000000005609228993004070 0.00000000000000560922899300407088 0.000000000000000828636101239238 0.000000000000000828636101239238

89 0.000000000000000111726215897425 0.00000000000000011172621589742590 0.000000000000000013655426387463 0.00000000000000001365542638746391 0.000000000000000001500596306315 0.00000000000000000150059630631592 0.000000000000000000146797464748 0.00000000000000000014679746474893 0.000000000000000000012627738903 0.00000000000000000001262773890394 0.000000000000000000000940363535 0.00000000000000000000094036353595 0.000000000000000000000059391381 0.00000000000000000000005939138196 0.000000000000000000000003093301 0.00000000000000000000000309330197 0.000000000000000000000000127559 0.000000000000000000000000127559


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98 0.000000000000000000000000003905 0.00000000000000000000000000390599 0.000000000000000000000000000079 0.000000000000000000000000000079

100 0.000000000000000000000000000001 0.000000000000000000000000000001


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E(X) 500

7.88861E-297.80972E-273.82676E-251.23732E-23

2.96957E-225.64218E-218.83942E-201.17438E-181.35054E-171.36554E-161.22899E-159.94363E-15

7.292E-144.88003E-132.99773E-12

1.69871E-118.91825E-114.35421E-101.98358E-098.45633E-093.38253E-081.27248E-074.51151E-071.51037E-064.78285E-061.43485E-054.08382E-050.0001104140.0002839230.0006951210.0016219480.0036101430.0076715540.0155755790.0302349480.0561506180.0998233210.169969439

0.2773185590.4337546680.6506320030.9362753211.2929516341.7139126312.1813433482.666086314

3.1297535

0.000000000000000000000000000000

0.010000000000000000000000000000

0.020000000000000000000000000000

0.030000000000000000000000000000

0.040000000000000000000000000000

0.050000000000000000000000000000

0.060000000000000000000000000000

0.070000000000000000000000000000

0.080000000000000000000000000000

0.090000000000000000000000000000

0 3 6 9 12 15 18 21 24 27 30

Discrete Probab


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3.52929653.8234045413.9794618693.9794618693.823404541

3.52929653.1297535

2.6660863142.1813433481.7139126311.2929516340.9362753210.6506320030.4337546680.2773185590.1699694390.0998233210.056150618

0.0302349480.0155755790.0076715540.0036101430.0016219480.0006951210.0002839230.0001104144.08382E-051.43485E-054.78285E-061.51037E-064.51151E-071.27248E-073.38253E-088.45633E-091.98358E-094.35421E-108.91825E-111.69871E-112.99773E-124.88003E-13

7.292E-14

9.94363E-151.22899E-151.36554E-161.35054E-171.17438E-188.83942E-205.64218E-212.96957E-221.23732E-23


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3.82676E-257.80972E-277.88861E-29


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N 100 TOTAL BASKETP(MAKE) 0.4P(LOSE) 0.6

PROBABILITIES 1K P(OF ' K 'Basket) Combination P(X=K) E(X) 40

0 6.53319E-23 1 6.5332E-23 01 4.35546E-23 100 4.3555E-21 4.36E-21

2 2.90364E-23 4950 1.4373E-19 2.87E-193 1.93576E-23 161700 3.1301E-18 9.39E-184 1.29051E-23 3921225 5.0604E-17 2.02E-165 8.60337E-24 75287520 6.4773E-16 3.24E-156 5.73558E-24 1192052400 6.8371E-15 4.1E-147 3.82372E-24 16007560800 6.1208E-14 4.28E-138 2.54915E-24 1.86088E+11 4.7437E-13 3.79E-129 1.69943E-24 1.90223E+12 3.2327E-12 2.91E-11

10 1.13295E-24 1.73103E+13 1.9612E-11 1.96E-1011 7.55303E-25 1.4163E+14 1.0697E-10 1.18E-0912 5.03535E-25 1.05042E+15 5.2892E-10 6.35E-09

13 3.3569E-25 7.11054E+15 2.3869E-09 3.1E-0814 2.23793E-25 4.41869E+16 9.8887E-09 1.38E-0715 1.49196E-25 2.53338E+17 3.7797E-08 5.67E-0716 9.94638E-26 1.34586E+18 1.3386E-07 2.14E-0617 6.63092E-26 6.65013E+18 4.4096E-07 7.5E-0618 4.42061E-26 3.06645E+19 1.3556E-06 2.44E-0519 2.94707E-26 1.32342E+20 3.9002E-06 7.41E-0520 1.96472E-26 5.35983E+20 1.0531E-05 0.00021121 1.30981E-26 2.04184E+21 2.6744E-05 0.00056222 8.73207E-27 7.33207E+21 6.4024E-05 0.00140923 5.82138E-27 2.48653E+22 0.00014475 0.00332924 3.88092E-27 7.97761E+22 0.0003096 0.00743125 2.58728E-27 2.42519E+23 0.00062747 0.01568726 1.72485E-27 6.99575E+23 0.00120666 0.03137327 1.1499E-27 1.91735E+24 0.00220477 0.05952928 7.66602E-28 4.99881E+24 0.0038321 0.10729929 5.11068E-28 1.24108E+25 0.00634278 0.18394130 3.40712E-28 2.93723E+25 0.0100075 0.30022531 2.27141E-28 6.63246E+25 0.01506506 0.46701732 1.51427E-28 1.43013E+26 0.02165603 0.69299333 1.00952E-28 2.94692E+26 0.02974969 0.9817434 6.73011E-29 5.80717E+26 0.03908293 1.3288235 4.48674E-29 1.09507E+27 0.04913282 1.719649

36 2.99116E-29 1.9772E+27 0.05914136 2.12908937 1.99411E-29 3.42003E+27 0.06819905 2.52336538 1.3294E-29 5.67005E+27 0.0753779 2.8643639 8.8627E-30 9.01392E+27 0.07988768 3.1156240 5.90847E-30 1.37462E+28 0.08121914 3.24876641 3.93898E-30 2.01164E+28 0.07923819 3.24876642 2.62598E-30 2.82588E+28 0.07420719 3.11670243 1.75066E-30 3.81165E+28 0.06672895 2.86934544 1.1671E-30 4.93782E+28 0.05762955 2.5357

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 2 4 6


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45 7.78069E-31 6.14485E+28 0.04781118 2.15150346 5.18713E-31 7.3471E+28 0.03811036 1.75307747 3.45809E-31 8.44135E+28 0.02919091 1.37197348 2.30539E-31 9.32066E+28 0.02148776 1.03141249 1.53693E-31 9.89131E+28 0.01520222 0.74490950 1.02462E-31 1.00891E+29 0.01033751 0.51687651 6.83079E-32 9.89131E+28 0.00675654 0.344584

52 4.55386E-32 9.32066E+28 0.0042445 0.22071453 3.03591E-32 8.44135E+28 0.00256271 0.13582454 2.02394E-32 7.3471E+28 0.00148701 0.08029855 1.34929E-32 6.14485E+28 0.00082912 0.04560256 8.99528E-33 4.93782E+28 0.00044417 0.02487457 5.99685E-33 3.81165E+28 0.00022858 0.01302958 3.9979E-33 2.82588E+28 0.00011298 0.00655359 2.66527E-33 2.01164E+28 5.3616E-05 0.00316360 1.77684E-33 1.37462E+28 2.4425E-05 0.00146561 1.18456E-33 9.01392E+27 1.0678E-05 0.00065162 7.89709E-34 5.67005E+27 4.4777E-06 0.000278

63 5.26473E-34 3.42003E+27 1.8006E-06 0.00011364 3.50982E-34 1.9772E+27 6.9396E-07 4.44E-0565 2.33988E-34 1.09507E+27 2.5623E-07 1.67E-0566 1.55992E-34 5.80717E+26 9.0587E-08 5.98E-0667 1.03995E-34 2.94692E+26 3.0646E-08 2.05E-0668 6.93297E-35 1.43013E+26 9.915E-09 6.74E-0769 4.62198E-35 6.63246E+25 3.0655E-09 2.12E-0770 3.08132E-35 2.93723E+25 9.0506E-10 6.34E-0871 2.05421E-35 1.24108E+25 2.5495E-10 1.81E-0872 1.36948E-35 4.99881E+24 6.8458E-11 4.93E-0973 9.12984E-36 1.91735E+24 1.7505E-11 1.28E-0974 6.08656E-36 6.99575E+23 4.258E-12 3.15E-1075 4.05771E-36 2.42519E+23 9.8407E-13 7.38E-1176 2.70514E-36 7.97761E+22 2.1581E-13 1.64E-1177 1.80343E-36 2.48653E+22 4.4843E-14 3.45E-1278 1.20228E-36 7.33207E+21 8.8152E-15 6.88E-1379 8.01522E-37 2.04184E+21 1.6366E-15 1.29E-1380 5.34348E-37 5.35983E+20 2.864E-16 2.29E-1481 3.56232E-37 1.32342E+20 4.7144E-17 3.82E-1582 2.37488E-37 3.06645E+19 7.2825E-18 5.97E-1683 1.58325E-37 6.65013E+18 1.0529E-18 8.74E-1784 1.0555E-37 1.34586E+18 1.4206E-19 1.19E-1785 7.03668E-38 2.53338E+17 1.7827E-20 1.52E-18

86 4.69112E-38 4.41869E+16 2.0729E-21 1.78E-1987 3.12741E-38 7.11054E+15 2.2238E-22 1.93E-2088 2.08494E-38 1.05042E+15 2.1901E-23 1.93E-2189 1.38996E-38 1.4163E+14 1.9686E-24 1.75E-2290 9.26641E-39 1.73103E+13 1.604E-25 1.44E-2391 6.17761E-39 1.90223E+12 1.1751E-26 1.07E-2492 4.11841E-39 1.86088E+11 7.6639E-28 7.05E-2693 2.7456E-39 16007560800 4.395E-29 4.09E-2794 1.8304E-39 1192052400 2.1819E-30 2.05E-28


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95 1.22027E-39 75287520 9.1871E-32 8.73E-3096 8.13512E-40 3921225 3.19E-33 3.06E-3197 5.42342E-40 161700 8.7697E-35 8.51E-3398 3.61561E-40 4950 1.7897E-36 1.75E-3499 2.41041E-40 100 2.4104E-38 2.39E-36

100 1.60694E-40 1 1.6069E-40 1.61E-38


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8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66

P(X=K)


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68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100

P(X=K)


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E(X)= 2K 1

P(X=K) 0.270670566IN % 27.06705665

Occurrence Applications of the Poisson distribution can be found in many fields related to counting:[22]

Telecommunication example: telephone calls arriving in a system.Astronomy example: photons arriving at a telescope.Biology example: the number of mutations on a strand of DNA per unit length.Management example: customers arriving at a counter or call centre.Civil engineering example: cars arriving at a traffic light.Finance and insurance example: number of Losses/Claims occurring in a given period of Time.Earthquake seismology example: an asymptotic Poisson model of seismic risk for large earthquakes. (LomnitRadioactivity example: Decay of a radioactive nucleus.

The Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of disThe number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry. This example waThe number of yeast cells used when brewing Guinness beer. This example was made famous by William SeThe number of phone calls arriving at a call centre within a minute. This example was made famous by A.K.Internet traffic.The number of goals in sports involving two competing teams.The number of deaths per year in a given age group.The number of jumps in a stock price in a given time interval.Under an assumption of homogeneity, the number of times a web server is accessed per minute.The number of mutations in a given stretch of DNA after a certain amount of radiation.The proportion of cells that will be infected at a given multiplicity of infection.The arrival of photons on a pixel circuit at a given illumination and over a given time period.The targeting of V-1 flying bombs on London during World War II.[24]Gallagher in 1976 showed that the counts of prime numbers in short intervals obey a Poisson distribution pr


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z, 1994).

rete properties (that is, those that may happen 0, 1, 2, 3, ... times during a given period of time or in a gis made famous by a book of Ladislaus Josephovich Bortkiewicz(18681931).

aly Gosset (18761937).*23+rlang (1878 1929).

ovided a certain version of an unproved conjecture of Hardy and Littlewood is true.[25]


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ven area) whenever the probability of the phenomenon happening is constant in time or space. Example


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of events that may be modelled as a Poisson distribution include:


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X' YEARS SINCE 1995 INCOME y = 1882.3x + 528471995 0 53807 528471996 1 55217 54729.31997 2 55209 56611.61998 3 55415 58493.91999 4 63100 60376.2

2000 5 63206 62258.52001 6 63761 64140.82002 7 65766 66023.12003 8 67905.42004 9 69787.72005 10 716702006 11 73552.32007 12 75434.62008 13 77316.92009 14 79199.22010 15 81081.5

0

10000

20000

30000

40000

50000

60000

70000

0 1 2 3 4 5

I N C O M E

Axis Title

SCATTER PLOT AND CHART LAYO


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y = 1882.3x + 52847R = 0.8756

6 7 8

UT 9

INCOME

Linear (INCOME)


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x y Y*4 3 3.6666671 2 0.738095-1 -1 -1.21429-2 -3 -2.19048

m 0.97619

b -0.2381

y = 0.9762x - 0.2381R = 0.8796

88% of total variation in (Y from Mean) is expalined by variation in x ( or regression line)

-3 -2

y


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sqrd err W/L sqrd err from Mean Y0.444444444 7.56251.592403628 3.06250.045918367 1.56250.655328798 10.5625

2.738095238 22.75

R^2 0.879644165

y = 0.9762x - 0.2381R = 0.8796

-4

-3

-2

-1

0

1

2

3

4

-1 0 1 2 3 4 5

x

Regression line

y

Linear (y)

Probability Examples of Discrete Probability Distribution

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