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Poisson Distribution

This page provides comprehensive tables for the Poisson Distribution. These tables are a fundamental statistical tool used to model the probability of a specific number of events occurring within a fixed interval of time or space, provided these events happen with a known average rate and independently of the time since the last event. The Poisson distribution is a cornerstone for analyzing count data where events are relatively rare but occur over a large observational space or time.

The Poisson distribution models discrete events that satisfy specific criteria: the events must occur independently of one another, and they must occur at a constant average rate over the interval. Examples of phenomena that can often be modeled using the Poisson distribution include the number of typos on a page, the number of calls arriving at a call center per hour, the number of defective items in a batch of manufactured goods, or the number of radioactive decays per second. The crucial parameter of the Poisson distribution is the average rate of occurrence, denoted by the Greek letter $\lambda$ (lambda).

The tables presented here list the probabilities of observing exactly 'k' events in the specified interval, given a specific average rate $\lambda$. That is, for a random variable $X$ following a Poisson distribution with rate $\lambda$, these tables provide the value of the probability mass function $P(X=k)$. The tables are typically structured with different values of the average rate $\lambda$ displayed across columns or rows, and different possible numbers of occurrences 'k' listed along the other axis.

Using the table is straightforward: you identify the column or section corresponding to your specific average rate $\lambda$ and then find the row corresponding to the desired number of events 'k'. The value at the intersection of that row and column is the probability $P(X=k)$ – the likelihood of observing exactly 'k' events. This value is calculated using the Poisson probability formula: $$ P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} $$ Where $e$ is the base of the natural logarithm (approximately $e \approx 2.71828$), $\lambda$ is the average rate, $k$ is the number of occurrences, and $k!$ is the factorial of $k$ ($k! = k \times (k-1) \times \dots \times 2 \times 1$).

Before the widespread availability of statistical software and calculators, these tables were indispensable for applying the Poisson distribution in practical scenarios. They eliminate the need for manual calculation of the formula involving powers of $\lambda$, the value of $e$ raised to the power of $-\lambda$, and factorials, which can be computationally intensive.

Poisson distribution tables are valuable tools in various fields:

By providing direct access to the pre-calculated probabilities $P(X=k)$ for various $\lambda$ and $k$, this resource allows for quick and accurate assessment of the likelihood of a specific number of random events occurring, significantly simplifying the application of the Poisson distribution.

$$P(x) = \frac{e^{-\lambda} \lambda^x}{x!}$$

For a given value of λ an entry indicates the probability of a specific value of x.

$$\lambda$$
$x$ 1 2 3 4 5 6 7 8 9 10
0 0.3679 0.1353 0.0498 0.0183 0.0067 0.0025 0.0009 0.0003 0.0001 0.0000
1 0.3679 0.2707 0.0733 0.0337 0.0149 0.0064 0.0027 0.0027 0.0011 0.0005
2 0.1839 0.2707 0.2240 0.1465 0.0842 0.0446 0.0223 0.0107 0.0050 0.0023
3 0.0613 0.1804 0.1954 0.1404 0.0892 0.0521 0.0286 0.0152 0.0150 0.0076
4 0.0153 0.0902 0.1680 0.1954 0.1755 0.1339 0.912 0.05137 0.337 0.0189
5 0.0031 0.0361 0.1008 0.1563 0.1755 0.1606 0.1277 0.0916 0.0607 0.0378
6 0.005 0.120 0.0504 0.1042 0.1462 0.1606 0.1490 0.1221 0.0911 0.0631
7 0.0001 0.0034 0.216 0.595 0.1044 0.1377 0.1490 0.1396 0.1171 0.09601
8 0.0000 0.0009 0.0081 0.0298 0.0653 0.1033 0.13047 0.1396 0.1318 0.1126
9 0.0002 0.0027 0.0132 0.0363 0.688 0.1014 0.1241 0.1318 0.1251
10 0.0000 0.0008 0.0053 0.0181 0.0413 0.0710 0.0993 0.1186 0.1241
11 0.0002 0.0019 0.0082 0.0225 0.0452 0.0722 0.0970 0.1137
12 0.0001 0.006 0.0034 0.0113 0.0263 0.0481 0.0728 0.0948
13 0.0000 0.0002 0.0013 0.0052 0.0142 0.0296 0.0504 0.0729
14 0.0001 0.0005 0.0022 0.0071 0.0169 0.0324 0.0521
15 0.0000 0.0002 0.0009 0.0033 0.0090 0.0194 0.0347
16 0.0000 0.0003 0.0014 0.0045 0.0109 0.0217
17 0.0001 0.0006 0.0021 0.0058 0.128
18 0.0000 0.0002 0.0009 0.0029 0.0071
19 0.0001 0.0004 0.0014 0.0037
20 0.0000 0.0002 0.0006 0.0019
21 0.0001 0.0003 0.0009
22 0.0000 0.0001 0.0004
23 .0000 .0002
24 0.0001
25 0.0000
$$\lambda$$
$x$ 11 12 13 14 15 16 17 18 19 20
0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
1 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
2 0.0010 0.004 0.002 0.0010 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
3 0.0037 0.0018 0.0008 0.0004 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000
4 0.0102 0.0053 0.0027 0.0013 0.0006 0.0003 0.0001 0.0001 0.0000 0.0000
5 0.0224 0.0127 0.0070 0.0037 0.0019 0.0010 0.0005 0.0002 0.0001 0.0000
6 0.0411 0.0255 0.0152 0.0087 0.0048 0.0026 0.0014 0.0007 0.0004 0.0002
7 0.0646 0.0437 0.0281 0.174 0.0104 0.0060 0.0034 0.0019 0.0010 0.0005
8 0.0888 0.0655 0.0457 0.0304 0.0194 0.120 0.0072 0.0042 0.0024 0.0013
9 0.01085 0.0874 0.0661 0.0473 0.0213 0.0135 0.0083 0.0050 0.0050 0.0029
10 0.1194 0.1048 0.0663 0.0486 0.0341 0.0230 0.0150 .0.0150 0.0095 0.0058
11 0.1194 0.1144 0.0844 0.0663 0.0496 0.0355 0.0245 0.0164 0.0164 .0.0106
12 0.1094 0.1144 0.1099 0.0984 0.0829 0.0661 0.0504 0.0368 0.0259 0.0176
13 0.0926 0.1056 0.1099 0.1060 0.0956 0.0814 0.0658 0.0509 0.0378 0.0271
14 0.0728 0.0905 0.1021 0.1021 0.1060 0.0930 0.0800 0.0655 0.0514 0.0387
15 0.0.534 0.0724 0.0885 0.0989 0.1024 0.0992 0.0906 0.0786 0.0650 0.0516
16 0.0367 0.0543 0.0719 0.0866 0.0960 0.0992 0.0963 0.0884 0.0772 0.0646
17 0.0237 0.0383 0.0550 0.0713 0.0847 0.0934 0.0963 0.0936 0.0863 0.0760
18 0.0145 0.0255 0.0397 0.0554 0.0706 0.0830 0.0909 0.0936 0.0911 0.0844
19 0.0084 0.0161 0.0272 0.0409 0.0557 0.0699 0.0814 0.0887 0.0911 0.0888
20 0.0046 0.0097 0.0177 0.0418 0.0559 0.0692 0.0798 0.0866 0.0866 0.0888
21 0.0024 0.0055 0.0109 0.0191 0.0209 0.0426 0.0560 0.0684 0.0783 0.0846
22 0.0012 0.0030 0.0065 0.0121 0.0204 0.0310 0.0433 0.0560 0.0676 0.0769
23 0.0006 0.0016 0.0037 0.0074 0.0133 0.0310 0.0216 0.0538 0.0559 0.00669
24 0.0003 0.0008 0.0020 0.0043 0.0083 0.0144 0.02226 0.0328 0.0442 0.0557
25 0.0001 0.0004 0.0010 0.0024 0.0050 0.0092 0.0124 0.0237 0.0336 0.0446