Question

Snowfall in the City of Cleveland for the winter months (defined as January, February, and March) is distributed as a Poisson random variable, with an average of 7 inches per month What is the probability that there will be no more than 4 Inches of snow in the City of Cleveland next January? ANSWER: (Use only the appropriate formula and/or statistical table in your textbook to answer this question. Report your answer as to 4 decimal places, using conventional rounding rules. What is the probability that there will be between 9 and 12 inches of snow in the City of Cleveland next March? ANSWER (Use only the appropriate formula and/or statistical table in your textbook to answer this question. Report your answer as to 4 decimal places, using conventional rounding rules. What is the probability that there will be exactly 8 inches of snow in the City of Cleveland next January and February combined? ANSWER: (Use only the appropriate formule and/or statistical table in your textbook to answer this question. Report your answer as to 4 decimal places, using conventional rounding rules

Answer 1

Let X denote the Poisson random variable with an average of 7 inches per month. Given, , the probability mass function is expressed as:

XP(7)

, x = 0,1,2....

P(X = 1) = !

1. To find:

P(X < 4) =?

P(X<4)

= P(X=0) + P(X= 1) + P(X = 2) + P(X = 3) + P(X = 4)

We can either substitute the value of x for each term and sum them up or we may use a cumulative poisson distribution table as the one given below:

For

= 7,0 = 4

= | 6. 0 6 .5 | 7.0 | 7.5 8. 0 8 .5 9. 0 9 .5 10.0 11.0 10.0 12.0 14.0 15.0 || 10.0025 0.0015 0.0009| 0.0006 0.0003 0.0002 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 10.0174 0.0113| 0.0073| 0.0047 0.0030 0.0019 0.0012 0.0008 0.0005 0.0002 0.0005 0.0001 0.0000 0.0000 20.0620 0.0430 0.0296| 0.0203 0.0138 0.0093 0.0062 0.0042 0.0028 0.0012 0.0028 0.0005 0.0001 0.0000 30.1512 0.1118| 0.0818| 0.0591 0.0424 0.0301 0.0212 0.0149 0.0103 0.0049 0.0103 0.0023 0.0005 0.0002 |4|0.2851 0.223710.1730] 0.1321 0.0996 0.0744 0.0550 0.0403 0.0293 0.0151 0.0293 0.0076 0.0018 0.0009 510.4457 0.3690| 0.3007] 0.2414 0.1912 0.1496 0.1157 0.0885 0.0671 0.0375 0.0671 0.0203 0.0055 0.0028 610.6063 0.5265 0.4497| 0.3782 0.3134 0.2562 0.2068 0.1649 0.1301 0.0786 0.1301 0.0458 0.0142 0.0076 710.7440 0.6728| 10.5987] 0.5246 0.4530 0.3856 0.3239 0.2687 0.2202 0.1432 0.2202 0.0895 0.0316 0.0180 18 0.8472 0.7916 | | 0.7291 0.6620 0.5925 0.5231 0.4557 0.3918 0.3328 0.2320 0.3328 0.1550 0.0621 0.0374 9|0.9161 0.8774 | 0.8305 0.7764 0.7166 0.6530 0.5874 0.5218 0.4579 0.3405 0.4579 0.2424 0.1094 0.0699 10| 0.9574 0.9332| 0.9015| 0.8622 0.8159 0.7634 0.7060 0.6453 0.5830 0.4599 0.5830 0.3472 0.1757 0.1185 11| 0.9799 0.9661| 0.9467 0.9208 0.8881 0.8487 0.8030 0.7520 0.6968 0.5793 0.6968 0.4616 0.2600 0.1848 12| 0.9912 0.9840| 0.9730|| | 0.9573 0.9362 0.9091 0.8758 0.8364 0.7916 0.6887 0.7916 0.5760 0.3585 0.2676 13| 0.9964 0.9929 0.9872 | 0.9784 0.9658 0.9486 0.9261 0.8981 0.8645 0.7813 0.8645 0.6815 0.4644 0.3632 14 0.9986 0.9970 0.9943 | 0.9897 0.9827 0.9726 0.9585 0.9400 0.9165 0.8540 0.9165 0.7720 0.5704 0.4657 15] 0.9995 0.9988| 10.9976 | 0.9954 0.9918 0.9862 0.9780 0.9665 0.9513 0.9074 0.9513 0.8444 0.6694 0.5681 16] 0.9998 0.9996 | 0.9990| 0.9980 0.9963 0.9934 0.9889 0.9823 0.9730 0.9441 0.9730 0.8987 0.7559 0.6641 1710.9999 0.9998 | 0.9996 | 0.9992 0.9984 0.9970 0.9947 0.9911 0.9857 0.9678 0.9857 0.9370 0.8272 0.7489 18| 1.0000 0.9999| 0.9999| 0.9997 0.9993 0.9987 0.9976 0.9957 0.9928 0.9823 0.9928 0.9626 0.8826 0.8195 19] 1.0000 1.0000 | 1.0000 | 0.9999 0.9997 0.9995 0.9989 0.9980 0.9965 0.9907 0.9965 0.9787 0.9235 0.8752 20|1.0000 1.0000 | 1.0000|1.0000 0.9999 0.9998 0.9996 0.9991 0.9984 0.9953 0.9984 0.9884 0.9521 0.9170 linnon innnnlinannlinannnnnnnogg na99a0096 anagin00770003 n0039 n07170460

= 0.1730

Hence,

P(X< 4) = 0.1730

2. To find:

P9<x< 12) =?

P(9<x<12)

= P(X=10)+ P(X = 11

From poisson distribution table for exact probabilities:

A x 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 0.0009 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0.0022 0.0137 0.0417 0.0848 0.1294 0.1579 0.1605 0.1399 0.1066 0.0723 0.0441 0.0245 0.0124 0.0058 0.0025 0.0020 0.0126 0.0390 0.0806 0.1249 0.1549 0.1601 0.1418 0.1099 0.0757 0.0469 0.0265 0.0137 0.0065 0.0029 0.0018 0.0116 0.0364 0.0765 0.1205 0.1519 0.1595 0.1435 0.1130 0.0791 0.0498 0.0285 0.0150 0.0073 0.0033 0.0017 0.0106 0.0340 0.0726 0.1162 0.1487 0.1586 0.1450 0.1160 0.0825 0.0528 0.0307 0.0164 0.0081 0.0037 0.0015 0.0014 0.0012 0.0098 0.0090 0.0082 0.0318 0.0296 0.0276 0.0688 0.06520.0617 0.1118 0.1076 0.1034 0.1454 0.1420 0.1385 0.1575 0.1562 0.1546 0.1462 0.1472 0.1480 0.1188 0.1215 0.1240 0.0858 0.0891 0.0923 0.0558 0.0588 0.0618 0.0330 0.0353 0.0377 0.0179 0.0194 0.0210 0.00890.0098 0.0108 0.0041 0.0046 0.0052 0.0011 0.0076 0.0258 0.0584 0.0992 0.1349 0.1529 0.1486 0.1263 0.0954 0.0649 0.0401 0.0277 0.0119 0.0058 0.0010 0.0070 0.0240 0.0552 0.0952 0.1314 0.1511 0.1489 0.1284 0.0985 0.0679 0.0426 0.0245 0.0130 0.0064 0.0064 0.0223 0.0521 0.0912 0.1277 0.1490 0.1490 0.1304 0.1014 0.0710 0.0452 0.0264 0.0142 0.0071

= 0.0710 + 0.0452

= 0.1162

Hence,

= P(9<x<12) = 0.1162

3. To find:

P(X= 8) =?

| x 0 1 2 3 4 5 6 7 18 9 10 II 12 13 14 6.1 0.0022 0.0137 0.0417 0.0848 0.1294 0.1579 0.1605 0.139 0.1066 0.0723 0.0441 0.0245 0.0124 0.0058 0.0025 6.2 0.0020 0.0126 0.0390 0.0806 0.1249 0.1549 0.1601 0.1418 0.1099 0.0757 0.0469 0.0265 0.0137 0.0065 0.0029 6.3 0.0018 0.0116 0.0364 0.0765 0.1205 0.1519 0.1595 0.1435 0.1130 0.0791 0.0498 0.0285 0.0150 0.0073 0.0033 6.4 0.0017 0.0106 0.0340 0.0726 0.1162 0.1487 0.1586 0.1450 0.1160 0.0825 0.0528 0.0307 0.014 0.0081 0.0037 6.5 0.0015 0.0098 0.0318 0.0688 0.1118 0.1454 0.1575 0.1462 0.118 0.0858 0.0558 0.0330 0.0179 0.0089 0.0041 6.6 0.0014 0.0090 0.0296 0.0652 0.1076 0.1420 0.1562 0.1472 0.1215 0.0891 0.0588 0.0353 0.014 0.0098 0.0046 6.7 0.0012 0.0082 0.0276 0.0617 0.1034 0.1385 0.1546 0.1480 0.1240 0.0923 0.0618 0.0377 0.0210 0.0108 0.0052 6.8 0.0011 0.0076 0.0258 0.0584 0.0992 0.1349 0.1529 0.1486 0.1263 0.0954 0.0649 0.0401 0.0277 0.0119 0.0058 6.9 17.0 0.0010 0.0009 0.0070 0.0064 0.0240 | 0.0223 0.0552 | 0.0521 0.0952 | 0.0912 0.1314 | 0.1277 0.1511 | 0.1490 0.1489 10.1490 0.1284 | 0.1304 0.0985 | 0.1014 0.0679 | 0.0710 0.0426 | 0.0452 0.0245 | 0.0264 0.0130 | 0.0142 0.0064 | 0.0071

$P(X=8)$ = 0.1304