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Cumulative Poisson Distribution

This page features detailed tables for the Cumulative Poisson Distribution. These tables serve a distinct but related purpose compared to standard Poisson probability tables, which provide the likelihood of observing exactly 'k' events. Instead, these tables are designed to give you the probability of observing 'k' or fewer events within a fixed interval of time or space, given the known average rate of event occurrence. This cumulative probability is denoted as $P(X \le k)$.

The Cumulative Poisson Distribution is particularly useful for answering practical questions that involve thresholds, such as "What is the probability that the number of arrivals will not exceed 10?" or "What is the chance of having 5 or fewer defects in this sample?". Like the standard Poisson distribution, it applies to discrete events that occur independently and at a constant average rate ($\lambda$, lambda) over the defined interval.

The structure of the Cumulative Poisson Distribution tables is typically organized to facilitate easy lookup. You will find different values of the average rate $\lambda$ represented, often in columns or distinct sections of the table. The rows usually correspond to various possible numbers of occurrences, 'k'. To find the cumulative probability $P(X \le k)$ for your specific scenario, you navigate the table to the intersection of the row corresponding to your desired maximum number of events 'k' and the column or section representing the average rate $\lambda$ for your context.

The value located at this intersection is the pre-calculated cumulative probability. This value is mathematically equivalent to the sum of the individual probabilities of observing exactly 0 events, exactly 1 event, exactly 2 events, and so on, up to exactly 'k' events. Using the formula for the individual Poisson probability $P(X=i) = \frac{e^{-\lambda} \lambda^i}{i!}$, the cumulative probability $P(X \le k)$ is the sum: $$ P(X \le k) = \sum_{i=0}^{k} P(X=i) = \sum_{i=0}^{k} \frac{e^{-\lambda} \lambda^i}{i!} $$ Calculating this sum manually, especially for larger values of 'k', can be very time-consuming and prone to errors.

Before the advent of modern statistical software and high-powered calculators, these tables were indispensable tools. They allowed statisticians, analysts, and researchers to quickly obtain cumulative probabilities without performing the extensive summation shown above for each required value of 'k' and $\lambda$. They significantly simplified the application of the Poisson distribution in real-world problems.

Cumulative Poisson tables are widely used in various analytical and decision-making contexts:

By providing these summed probabilities directly, this resource allows for rapid and accurate assessment of the likelihood that the number of events in a Poisson process will not exceed a specific upper bound, making it a highly practical tool for planning and analysis.

For a given value of λ an entry indicates the probability of a equal to or less than the 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 0.7358 0.4060 0.1991 0.0404 0.0174 0.0073 .0.0030 0.0012 0.0012 0.0005
2 0.9197 0.6767 0.4232 0.2381 0.1247 0.0620 0.0296 0.0138 0.0062 0.0028
3 0.9810 0.8571 0.6472 0.4335 0.2650 0.1512 0.0818 0.424 0.0212 0.0103
4 0.9963 0.9473 0.5153 0.6288 0.4405 0.2851 0.1730 0.0996 0.0550 0.0293
5 0.9994 0.9834 0.9161 0.7851 0.6160 0.4457 0.3007 0.1912 0.1157 0.0671
6 0.9999 0.9955 0.9665 0.8893 0.7622 0.6063 0.4497 0.3134 0.2068 0.1301
7 1.00000 0.9989 0.9881 0.9489 0.8666 0.7440 0.5987 0.4530 0.3239 0.2202
8 0.9998 0.9962 0.9786 0.9319 0.8472 0.7291 0.5925 0.4557 0.3328
9 1.0000 0.9989 0.9919 0.9682 0.9161 0.8305 0.7166 0.58740 0.4579
10 0.9997 0.9972 0.9863 0.9574 0.9015 0.8159 0.7060 0.5830
11 0.9999 0.9991 0.9945 0.9799 0.9467 0.8881 0.8030 0.6968
12 1.0000 0.9997 0.9980 0.9912 0.9730 0.9362 0.8758 0.7916
13 0.9999 0.9993 0.9964 0.9872 0.9658 0.9261 0.8645
14 1.0000 0.9998 0.9986 0.9943 0.9827 0.9585 0.9165
15 0.9999 0.9995 0.9976 0.9918 .0.9780 0.9513
16 1.0000 0.9998 0.9990 0.9963 0.9889 0.97360
17 0.9999 0.9996 0.9984 0.9947 0.9857
18 1.0000 0.9999 0.9993 0.9976 0.9928
19 1.0000 0.9997 0.9989 0.9965
20 0.9999 0.9996 0.9984
21 1.0000 0.9998 0.9993
22 0.9999 0.9997
23 1.0000 0.9999
24 1.0000
$$\lambda$$
$x$ 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 0.0012 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
3 0.0049 0.0023 0.0011 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000
4 0.0151 0.0076 0.0037 0.0018 0.0009 0.0004 0.0002 0.0001 0.0000 0.0000
5 0.0375 0.0203 0.0107 0.0055 0.0028 0.0014 0.0007 0.0003 0.0002 0.0001
6 0.0786 0.0458 0.0259 0.0142 0.0076 0.0040 0.0021 0.0010 0.0005 0.0003
7 0.1432 0.0895 0.0540 0.0316 0.0180 0.0100 0.0054 0.0029 0.0015 0.0008
8 0.2320 0.1550 0.0998 0.0621 0.0374 0.0220 0.0126 0.0071 0.0039 0.0021
9 0.3405 0.2424 0.1658 0.1094 0.0699 0.0433 0.0261 0.0154 0.0089 0.0050
10 0.4599 0.3472 0.2517 0.1757 0.1185 0.0774 0.0491 0.0304 0.0183 0.0108
11 0.5793 0.4616 0.3532 0.2600 0.1848 0.1270 0.0847 0.0549 0.0347 0.0214
12 0.6887 0.5760 0.4631 0.3585 0.2676 0.1931 0.1350 0.0917 0.0606 0.0390
13 0.7813 0.6815 0.5730 0.4655 0.3632 0.2745 0.2009 0.1426 0.0984 0.0661
14 0.8540 0.7720 0.6651 0.5704 0.4657 0.3675 0.2808 0.2081 0.1497 0.1049
15 0.9074 0.8444 0.7636 0.6694 0.5681 0.4667 0.3715 0.2867 0.2148 0.1565
16 0.9441 0.8987 0.8355 0.7559 0.6641 0.5660 0.4677 0.3751 0.2920 0.2211
17 0.9678 0.9370 0.8905 0.8272 0.7489 0.6593 0.5640 0.4686 0.3784 0.2970
18 0.9823 0.9626 0.9302 0.8826 0.8195 0.7423 0.6550 0.5622 0.4695 0.3814
19 0.9907 0.9787 0.9573 0.9235 0.8752 0.8122 0.7363 0.6509 0.5606 0.4703
20 0.9953 0.9884 0.9750 0.9521 0.9170 0.8682 0.8055 0.7307 0.6472 0.5591
21 0.9977 0.9939 0.9859 0.9712 0.6496 0.9108 0.8615 0.7991 0.7255 0.6437
22 0.9990 0.9970 0.9924 0.9833 0.9673 0.9418 0.9047 0.8551 0.7931 0.7206
23 0.9995 0.9985 0.9960 0.9907 0.9805 0.9633 0.9367 0.8989 0.8490 0.7875
24 0.9998 0.9993 0.9980 0.9950 0.9888 0.9777 0.9594 0.9317 0.8933 0.8432
25 0.9999 0.9997 0.9990 0.9974 0.9838 0.9869 0.9748 .9554 0.9269 0.8878