Discrete Distributions

Poisson Distribution Table: P(X = x)

The probability of exactly x events when events occur independently at an average rate of λ per interval.

λ = 0.1 to 20
All non-negligible x
Calculator included

Poisson Table

Rows: number of events x · Columns: mean λ · Cells: P(X = x)

λ = 0.1 to 1

λ = 0.1 to 1
Mean (λ) → x ↓	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
0	0.9048	0.8187	0.7408	0.6703	0.6065	0.5488	0.4966	0.4493	0.4066	0.3679
1	0.0905	0.1637	0.2222	0.2681	0.3033	0.3293	0.3476	0.3595	0.3659	0.3679
2	0.0045	0.0164	0.0333	0.0536	0.0758	0.0988	0.1217	0.1438	0.1647	0.1839
3	0.0002	0.0011	0.0033	0.0072	0.0126	0.0198	0.0284	0.0383	0.0494	0.0613
4	0+	0.0001	0.0003	0.0007	0.0016	0.0030	0.0050	0.0077	0.0111	0.0153
5	0+	0+	0+	0.0001	0.0002	0.0004	0.0007	0.0012	0.0020	0.0031
6	0+	0+	0+	0+	0+	0+	0.0001	0.0002	0.0003	0.0005
7	0+	0+	0+	0+	0+	0+	0+	0+	0+	0.0001

λ = 1.5 to 6

λ = 1.5 to 6
Mean (λ) → x ↓	1.5	2	2.5	3	3.5	4	4.5	5	5.5	6
0	0.2231	0.1353	0.0821	0.0498	0.0302	0.0183	0.0111	0.0067	0.0041	0.0025
1	0.3347	0.2707	0.2052	0.1494	0.1057	0.0733	0.0500	0.0337	0.0225	0.0149
2	0.2510	0.2707	0.2565	0.2240	0.1850	0.1465	0.1125	0.0842	0.0618	0.0446
3	0.1255	0.1804	0.2138	0.2240	0.2158	0.1954	0.1687	0.1404	0.1133	0.0892
4	0.0471	0.0902	0.1336	0.1680	0.1888	0.1954	0.1898	0.1755	0.1558	0.1339
5	0.0141	0.0361	0.0668	0.1008	0.1322	0.1563	0.1708	0.1755	0.1714	0.1606
6	0.0035	0.0120	0.0278	0.0504	0.0771	0.1042	0.1281	0.1462	0.1571	0.1606
7	0.0008	0.0034	0.0099	0.0216	0.0385	0.0595	0.0824	0.1044	0.1234	0.1377
8	0.0001	0.0009	0.0031	0.0081	0.0169	0.0298	0.0463	0.0653	0.0849	0.1033
9	0+	0.0002	0.0009	0.0027	0.0066	0.0132	0.0232	0.0363	0.0519	0.0688
10	0+	0+	0.0002	0.0008	0.0023	0.0053	0.0104	0.0181	0.0285	0.0413
11	0+	0+	0+	0.0002	0.0007	0.0019	0.0043	0.0082	0.0143	0.0225
12	0+	0+	0+	0.0001	0.0002	0.0006	0.0016	0.0034	0.0065	0.0113
13	0+	0+	0+	0+	0.0001	0.0002	0.0006	0.0013	0.0028	0.0052
14	0+	0+	0+	0+	0+	0.0001	0.0002	0.0005	0.0011	0.0022
15	0+	0+	0+	0+	0+	0+	0.0001	0.0002	0.0004	0.0009
16	0+	0+	0+	0+	0+	0+	0+	0+	0.0001	0.0003
17	0+	0+	0+	0+	0+	0+	0+	0+	0+	0.0001
18	0+	0+	0+	0+	0+	0+	0+	0+	0+	0+

λ = 6.5 to 12

λ = 6.5 to 12
Mean (λ) → x ↓	6.5	7	7.5	8	8.5	9	9.5	10	11	12
0	0.0015	0.0009	0.0006	0.0003	0.0002	0.0001	0.0001	0+	0+	0+
1	0.0098	0.0064	0.0041	0.0027	0.0017	0.0011	0.0007	0.0005	0.0002	0.0001
2	0.0318	0.0223	0.0156	0.0107	0.0074	0.0050	0.0034	0.0023	0.0010	0.0004
3	0.0688	0.0521	0.0389	0.0286	0.0208	0.0150	0.0107	0.0076	0.0037	0.0018
4	0.1118	0.0912	0.0729	0.0573	0.0443	0.0337	0.0254	0.0189	0.0102	0.0053
5	0.1454	0.1277	0.1094	0.0916	0.0752	0.0607	0.0483	0.0378	0.0224	0.0127
6	0.1575	0.1490	0.1367	0.1221	0.1066	0.0911	0.0764	0.0631	0.0411	0.0255
7	0.1462	0.1490	0.1465	0.1396	0.1294	0.1171	0.1037	0.0901	0.0646	0.0437
8	0.1188	0.1304	0.1373	0.1396	0.1375	0.1318	0.1232	0.1126	0.0888	0.0655
9	0.0858	0.1014	0.1144	0.1241	0.1299	0.1318	0.1300	0.1251	0.1085	0.0874
10	0.0558	0.0710	0.0858	0.0993	0.1104	0.1186	0.1235	0.1251	0.1194	0.1048
11	0.0330	0.0452	0.0585	0.0722	0.0853	0.0970	0.1067	0.1137	0.1194	0.1144
12	0.0179	0.0263	0.0366	0.0481	0.0604	0.0728	0.0844	0.0948	0.1094	0.1144
13	0.0089	0.0142	0.0211	0.0296	0.0395	0.0504	0.0617	0.0729	0.0926	0.1056
14	0.0041	0.0071	0.0113	0.0169	0.0240	0.0324	0.0419	0.0521	0.0728	0.0905
15	0.0018	0.0033	0.0057	0.0090	0.0136	0.0194	0.0265	0.0347	0.0534	0.0724
16	0.0007	0.0014	0.0026	0.0045	0.0072	0.0109	0.0157	0.0217	0.0367	0.0543
17	0.0003	0.0006	0.0012	0.0021	0.0036	0.0058	0.0088	0.0128	0.0237	0.0383
18	0.0001	0.0002	0.0005	0.0009	0.0017	0.0029	0.0046	0.0071	0.0145	0.0255
19	0+	0.0001	0.0002	0.0004	0.0008	0.0014	0.0023	0.0037	0.0084	0.0161
20	0+	0+	0.0001	0.0002	0.0003	0.0006	0.0011	0.0019	0.0046	0.0097
21	0+	0+	0+	0.0001	0.0001	0.0003	0.0005	0.0009	0.0024	0.0055
22	0+	0+	0+	0+	0.0001	0.0001	0.0002	0.0004	0.0012	0.0030
23	0+	0+	0+	0+	0+	0+	0.0001	0.0002	0.0006	0.0016
24	0+	0+	0+	0+	0+	0+	0+	0.0001	0.0003	0.0008
25	0+	0+	0+	0+	0+	0+	0+	0+	0.0001	0.0004
26	0+	0+	0+	0+	0+	0+	0+	0+	0+	0.0002
27	0+	0+	0+	0+	0+	0+	0+	0+	0+	0.0001
28	0+	0+	0+	0+	0+	0+	0+	0+	0+	0+

λ = 13 to 20

λ = 13 to 20
Mean (λ) → x ↓	13	14	15	16	17	18	19	20
0	0+	0+	0+	0+	0+	0+	0+	0+
1	0+	0+	0+	0+	0+	0+	0+	0+
2	0.0002	0.0001	0+	0+	0+	0+	0+	0+
3	0.0008	0.0004	0.0002	0.0001	0+	0+	0+	0+
4	0.0027	0.0013	0.0006	0.0003	0.0001	0.0001	0+	0+
5	0.0070	0.0037	0.0019	0.0010	0.0005	0.0002	0.0001	0.0001
6	0.0152	0.0087	0.0048	0.0026	0.0014	0.0007	0.0004	0.0002
7	0.0281	0.0174	0.0104	0.0060	0.0034	0.0019	0.0010	0.0005
8	0.0457	0.0304	0.0194	0.0120	0.0072	0.0042	0.0024	0.0013
9	0.0661	0.0473	0.0324	0.0213	0.0135	0.0083	0.0050	0.0029
10	0.0859	0.0663	0.0486	0.0341	0.0230	0.0150	0.0095	0.0058
11	0.1015	0.0844	0.0663	0.0496	0.0355	0.0245	0.0164	0.0106
12	0.1099	0.0984	0.0829	0.0661	0.0504	0.0368	0.0259	0.0176
13	0.1099	0.1060	0.0956	0.0814	0.0658	0.0509	0.0378	0.0271
14	0.1021	0.1060	0.1024	0.0930	0.0800	0.0655	0.0514	0.0387
15	0.0885	0.0989	0.1024	0.0992	0.0906	0.0786	0.0650	0.0516
16	0.0719	0.0866	0.0960	0.0992	0.0963	0.0884	0.0772	0.0646
17	0.0550	0.0713	0.0847	0.0934	0.0963	0.0936	0.0863	0.0760
18	0.0397	0.0554	0.0706	0.0830	0.0909	0.0936	0.0911	0.0844
19	0.0272	0.0409	0.0557	0.0699	0.0814	0.0887	0.0911	0.0888
20	0.0177	0.0286	0.0418	0.0559	0.0692	0.0798	0.0866	0.0888
21	0.0109	0.0191	0.0299	0.0426	0.0560	0.0684	0.0783	0.0846
22	0.0065	0.0121	0.0204	0.0310	0.0433	0.0560	0.0676	0.0769
23	0.0037	0.0074	0.0133	0.0216	0.0320	0.0438	0.0559	0.0669
24	0.0020	0.0043	0.0083	0.0144	0.0226	0.0328	0.0442	0.0557
25	0.0010	0.0024	0.0050	0.0092	0.0154	0.0237	0.0336	0.0446
26	0.0005	0.0013	0.0029	0.0057	0.0101	0.0164	0.0246	0.0343
27	0.0002	0.0007	0.0016	0.0034	0.0063	0.0109	0.0173	0.0254
28	0.0001	0.0003	0.0009	0.0019	0.0038	0.0070	0.0117	0.0181
29	0.0001	0.0002	0.0004	0.0011	0.0023	0.0044	0.0077	0.0125
30	0+	0.0001	0.0002	0.0006	0.0013	0.0026	0.0049	0.0083
31	0+	0+	0.0001	0.0003	0.0007	0.0015	0.0030	0.0054
32	0+	0+	0.0001	0.0001	0.0004	0.0009	0.0018	0.0034
33	0+	0+	0+	0.0001	0.0002	0.0005	0.0010	0.0020
34	0+	0+	0+	0+	0.0001	0.0002	0.0006	0.0012
35	0+	0+	0+	0+	0+	0.0001	0.0003	0.0007
36	0+	0+	0+	0+	0+	0.0001	0.0002	0.0004
37	0+	0+	0+	0+	0+	0+	0.0001	0.0002
38	0+	0+	0+	0+	0+	0+	0+	0.0001
39	0+	0+	0+	0+	0+	0+	0+	0.0001
40	0+	0+	0+	0+	0+	0+	0+	0+

Tip: click any value to highlight its row and column.

How to read this table

Find the block that contains your mean λ.
Go to the row for x and the column for λ.
The cell is P(X = x). Add cells for a range of counts, or use the cumulative table.
Rows stop once every remaining probability in the block is below 0.00005.

Worked example

A site records an average of 2.5 adverse events per month. The chance of exactly 4 events next month is row 4, column λ = 2.5: 0.1336. The chance of a month with no events is 0.0821.

Click the underlined links to highlight the value in the table.

Calculator

For values between the rows and columns of the table. Results update as you type.

Formula and how the values were computed

P(X = x) = e^−λ λ^x / x!

Each cell is computed exactly from the Poisson formula and rounded to four decimals.

Frequently asked questions

When is the Poisson distribution appropriate?

For counts of events that occur independently at a constant average rate, such as infections per 1,000 patient-days or adverse events per month. The mean and the variance are both λ.

What if my counts are more variable than Poisson?

When the variance is clearly larger than the mean (overdispersion), a negative binomial model is usually more appropriate.

How does the Poisson relate to the binomial?

With many trials and a small probability of success, the binomial is well approximated by a Poisson distribution with λ = np.

Poisson Distribution Table: P(X = x)

Poisson Table

λ = 0.1 to 1

λ = 1.5 to 6

λ = 6.5 to 12

λ = 13 to 20

How to read this table

Worked example

Calculator

Formula and how the values were computed

Frequently asked questions

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