A statistician\'s teenage daughter withdraws a certain amount of money X from an

Question

A statistician's teenage daughter withdraws a certain amount of money X from an ATM every so often, using a method that is unknown to him: she randomly spins a circular wheel that is equally divided among four regions, each containing a specific dollar amount, as shown. Bank statements reveal that over the past n = 80 ATM transactions, $10 was withdrawn thirteen times, $20 sixteen times, $30 nineteen times, and $40 thirty-two times. For this sample, construct a relative frequency table, and calculate the average amount x withdrawn per transaction, and the variance s^2. Suppose this process continues indefinitely. Construct a probability table, and calculate the expected amount mu withdrawn per transaction, and the variance sigma^2. (Verify that, for this sample, s^2 and sigma^2 happen to be equal.)

Explanation / Answer

x f f*x f*x^2 Relative freq. 10 13 130 1300 0.1625 20 16 320 6400 0.2 30 19 570 17100 0.2375 40 32 1280 51200 0.4 Total 80 2300 76000 1 Mean 28.75 Var(s2) 123.4375 Using Poission distribution we can calculate probability table as: Poisson(=28.75) Probability Table x p µ=x*p (x^2)*p 0 3.26613E-13 0 0 1 9.39013E-12 9.39013E-12 9.39013E-12 2 1.34983E-10 2.69966E-10 5.39932E-10 3 1.29359E-09 3.88076E-09 1.16423E-08 4 9.29766E-09 3.71907E-08 1.48763E-07 5 5.34616E-08 2.67308E-07 1.33654E-06 6 2.5617E-07 1.53702E-06 9.22212E-06 7 1.05213E-06 7.36489E-06 5.15542E-05 8 3.78108E-06 3.02486E-05 0.000241989 9 1.20785E-05 0.000108706 0.000978355 10 3.47255E-05 0.000347255 0.003472555 11 9.076E-05 0.000998359 0.010981954 12 0.000217446 0.002609349 0.031312184 13 0.00048089 0.006251565 0.081270339 14 0.000987541 0.013825575 0.193558056 15 0.001892787 0.028391807 0.4258771 16 0.003401102 0.054417629 0.87068207 17 0.005751863 0.097781678 1.662288523 18 0.009187004 0.165366073 2.97658931 19 0.013901388 0.264126366 5.018400958 20 0.019983245 0.399664896 7.993297925 21 0.027358014 0.574518288 12.06488406 22 0.03575195 0.786542895 17.30394369 23 0.044689937 1.027868556 23.64097678 24 0.053534821 1.284835695 30.83605667 25 0.061565044 1.539126093 38.47815231 26 0.068076731 1.769995006 46.01987017 27 0.072489112 1.957206017 52.84456245 28 0.074430784 2.084061962 58.35373494 29 0.07378914 2.139885051 62.05666646 30 0.070714592 2.121437766 63.64313297 31 0.065582081 2.033044525 63.02438029 32 0.058921401 1.885484842 60.33551495 33 0.051333039 1.693990288 55.9016795 34 0.043406614 1.475824872 50.17804565 35 0.035655433 1.247940149 43.67790522 36 0.028474825 1.025093694 36.90337298 37 0.022125708 0.818651214 30.29009491 38 0.016739845 0.636114119 24.17233652 39 0.012340271 0.48127055 18.76955147 40 0.008869569 0.354782778 14.1913111 41 0.006219515 0.255000121 10.45500498 42 0.004257406 0.178811061 7.510064551 43 0.002846522 0.122400428 5.263218425 44 0.001859943 0.081837496 3.600849814 45 0.001188297 0.053473364 2.406301368 46 0.000742686 0.034163538 1.571522745 47 0.000454302 0.021352211 1.003553927 48 0.000272108 0.013061193 0.626937265 49 0.000159655 0.00782311 0.38333241 50 9.18018E-05 0.00459009 0.229504514 51 5.1751E-05 0.002639302 0.134604398 52 2.86123E-05 0.001487842 0.077367772 53 1.55208E-05 0.000822605 0.043598056 54 8.26341E-06 0.000446224 0.024096113 55 4.31951E-06 0.000237573 0.013066522 56 2.21761E-06 0.000124186 0.006954413 57 1.11853E-06 6.37562E-05 0.003634102 58 5.54444E-07 3.21577E-05 0.001865148 59 2.70174E-07 1.59403E-05 0.000940475 60 1.29458E-07 7.7675E-06 0.00046605 61 6.10152E-08 3.72192E-06 0.000227037 62 2.82933E-08 1.75419E-06 0.00010876 63 1.29116E-08 8.13433E-07 5.12463E-05 64 5.80015E-09 3.71209E-07 2.37574E-05 65 2.56545E-09 1.66754E-07 1.0839E-05 66 1.11753E-09 7.37567E-08 4.86794E-06 67 4.79535E-10 3.21289E-08 2.15263E-06 68 2.02745E-10 1.37866E-08 9.37491E-07 69 8.4477E-11 5.82891E-09 4.02195E-07 70 3.46959E-11 2.42871E-09 1.7001E-07 71 1.40494E-11 9.97507E-10 7.0823E-08 72 5.61E-12 4.0392E-10 2.90822E-08 73 2.20942E-12 1.61288E-10 1.1774E-08 74 8.58389E-13 6.35208E-11 4.70054E-09 75 3.29049E-13 2.46787E-11 1.8509E-09 76 1.24476E-13 9.46016E-12 7.18972E-10 77 4.64763E-14 3.57868E-12 2.75558E-10 78 1.71307E-14 1.33619E-12 1.04223E-10 79 6.23428E-15 4.92508E-13 3.89081E-11 80 2.24044E-15 1.79235E-13 1.43388E-11 28.75 855.3125 variance(2) 28.75

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