Discrete Distributions

Cumulative Poisson Distribution Table: P(X ≤ x)

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

λ = 0.1 to 20
All non-negligible x
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Cumulative 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	0.9953	0.9825	0.9631	0.9384	0.9098	0.8781	0.8442	0.8088	0.7725	0.7358
2	0.9998	0.9989	0.9964	0.9921	0.9856	0.9769	0.9659	0.9526	0.9371	0.9197
3	1−	0.9999	0.9997	0.9992	0.9982	0.9966	0.9942	0.9909	0.9865	0.9810
4	1−	1−	1−	0.9999	0.9998	0.9996	0.9992	0.9986	0.9977	0.9963
5	1−	1−	1−	1−	1−	1−	0.9999	0.9998	0.9997	0.9994
6	1−	1−	1−	1−	1−	1−	1−	1−	1−	0.9999
7	1−	1−	1−	1−	1−	1−	1−	1−	1−	1−

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	0.5578	0.4060	0.2873	0.1991	0.1359	0.0916	0.0611	0.0404	0.0266	0.0174
2	0.8088	0.6767	0.5438	0.4232	0.3208	0.2381	0.1736	0.1247	0.0884	0.0620
3	0.9344	0.8571	0.7576	0.6472	0.5366	0.4335	0.3423	0.2650	0.2017	0.1512
4	0.9814	0.9473	0.8912	0.8153	0.7254	0.6288	0.5321	0.4405	0.3575	0.2851
5	0.9955	0.9834	0.9580	0.9161	0.8576	0.7851	0.7029	0.6160	0.5289	0.4457
6	0.9991	0.9955	0.9858	0.9665	0.9347	0.8893	0.8311	0.7622	0.6860	0.6063
7	0.9998	0.9989	0.9958	0.9881	0.9733	0.9489	0.9134	0.8666	0.8095	0.7440
8	1−	0.9998	0.9989	0.9962	0.9901	0.9786	0.9597	0.9319	0.8944	0.8472
9	1−	1−	0.9997	0.9989	0.9967	0.9919	0.9829	0.9682	0.9462	0.9161
10	1−	1−	0.9999	0.9997	0.9990	0.9972	0.9933	0.9863	0.9747	0.9574
11	1−	1−	1−	0.9999	0.9997	0.9991	0.9976	0.9945	0.9890	0.9799
12	1−	1−	1−	1−	0.9999	0.9997	0.9992	0.9980	0.9955	0.9912
13	1−	1−	1−	1−	1−	0.9999	0.9997	0.9993	0.9983	0.9964
14	1−	1−	1−	1−	1−	1−	0.9999	0.9998	0.9994	0.9986
15	1−	1−	1−	1−	1−	1−	1−	0.9999	0.9998	0.9995
16	1−	1−	1−	1−	1−	1−	1−	1−	0.9999	0.9998
17	1−	1−	1−	1−	1−	1−	1−	1−	1−	0.9999
18	1−	1−	1−	1−	1−	1−	1−	1−	1−	1−

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	0.0113	0.0073	0.0047	0.0030	0.0019	0.0012	0.0008	0.0005	0.0002	0.0001
2	0.0430	0.0296	0.0203	0.0138	0.0093	0.0062	0.0042	0.0028	0.0012	0.0005
3	0.1118	0.0818	0.0591	0.0424	0.0301	0.0212	0.0149	0.0103	0.0049	0.0023
4	0.2237	0.1730	0.1321	0.0996	0.0744	0.0550	0.0403	0.0293	0.0151	0.0076
5	0.3690	0.3007	0.2414	0.1912	0.1496	0.1157	0.0885	0.0671	0.0375	0.0203
6	0.5265	0.4497	0.3782	0.3134	0.2562	0.2068	0.1649	0.1301	0.0786	0.0458
7	0.6728	0.5987	0.5246	0.4530	0.3856	0.3239	0.2687	0.2202	0.1432	0.0895
8	0.7916	0.7291	0.6620	0.5925	0.5231	0.4557	0.3918	0.3328	0.2320	0.1550
9	0.8774	0.8305	0.7764	0.7166	0.6530	0.5874	0.5218	0.4579	0.3405	0.2424
10	0.9332	0.9015	0.8622	0.8159	0.7634	0.7060	0.6453	0.5830	0.4599	0.3472
11	0.9661	0.9467	0.9208	0.8881	0.8487	0.8030	0.7520	0.6968	0.5793	0.4616
12	0.9840	0.9730	0.9573	0.9362	0.9091	0.8758	0.8364	0.7916	0.6887	0.5760
13	0.9929	0.9872	0.9784	0.9658	0.9486	0.9261	0.8981	0.8645	0.7813	0.6815
14	0.9970	0.9943	0.9897	0.9827	0.9726	0.9585	0.9400	0.9165	0.8540	0.7720
15	0.9988	0.9976	0.9954	0.9918	0.9862	0.9780	0.9665	0.9513	0.9074	0.8444
16	0.9996	0.9990	0.9980	0.9963	0.9934	0.9889	0.9823	0.9730	0.9441	0.8987
17	0.9998	0.9996	0.9992	0.9984	0.9970	0.9947	0.9911	0.9857	0.9678	0.9370
18	0.9999	0.9999	0.9997	0.9993	0.9987	0.9976	0.9957	0.9928	0.9823	0.9626
19	1−	1−	0.9999	0.9997	0.9995	0.9989	0.9980	0.9965	0.9907	0.9787
20	1−	1−	1−	0.9999	0.9998	0.9996	0.9991	0.9984	0.9953	0.9884
21	1−	1−	1−	1−	0.9999	0.9998	0.9996	0.9993	0.9977	0.9939
22	1−	1−	1−	1−	1−	0.9999	0.9999	0.9997	0.9990	0.9970
23	1−	1−	1−	1−	1−	1−	0.9999	0.9999	0.9995	0.9985
24	1−	1−	1−	1−	1−	1−	1−	1−	0.9998	0.9993
25	1−	1−	1−	1−	1−	1−	1−	1−	0.9999	0.9997
26	1−	1−	1−	1−	1−	1−	1−	1−	1−	0.9999
27	1−	1−	1−	1−	1−	1−	1−	1−	1−	0.9999
28	1−	1−	1−	1−	1−	1−	1−	1−	1−	1−

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	0.0011	0.0005	0.0002	0.0001	0+	0+	0+	0+
4	0.0037	0.0018	0.0009	0.0004	0.0002	0.0001	0+	0+
5	0.0107	0.0055	0.0028	0.0014	0.0007	0.0003	0.0002	0.0001
6	0.0259	0.0142	0.0076	0.0040	0.0021	0.0010	0.0005	0.0003
7	0.0540	0.0316	0.0180	0.0100	0.0054	0.0029	0.0015	0.0008
8	0.0998	0.0621	0.0374	0.0220	0.0126	0.0071	0.0039	0.0021
9	0.1658	0.1094	0.0699	0.0433	0.0261	0.0154	0.0089	0.0050
10	0.2517	0.1757	0.1185	0.0774	0.0491	0.0304	0.0183	0.0108
11	0.3532	0.2600	0.1848	0.1270	0.0847	0.0549	0.0347	0.0214
12	0.4631	0.3585	0.2676	0.1931	0.1350	0.0917	0.0606	0.0390
13	0.5730	0.4644	0.3632	0.2745	0.2009	0.1426	0.0984	0.0661
14	0.6751	0.5704	0.4657	0.3675	0.2808	0.2081	0.1497	0.1049
15	0.7636	0.6694	0.5681	0.4667	0.3715	0.2867	0.2148	0.1565
16	0.8355	0.7559	0.6641	0.5660	0.4677	0.3751	0.2920	0.2211
17	0.8905	0.8272	0.7489	0.6593	0.5640	0.4686	0.3784	0.2970
18	0.9302	0.8826	0.8195	0.7423	0.6550	0.5622	0.4695	0.3814
19	0.9573	0.9235	0.8752	0.8122	0.7363	0.6509	0.5606	0.4703
20	0.9750	0.9521	0.9170	0.8682	0.8055	0.7307	0.6472	0.5591
21	0.9859	0.9712	0.9469	0.9108	0.8615	0.7991	0.7255	0.6437
22	0.9924	0.9833	0.9673	0.9418	0.9047	0.8551	0.7931	0.7206
23	0.9960	0.9907	0.9805	0.9633	0.9367	0.8989	0.8490	0.7875
24	0.9980	0.9950	0.9888	0.9777	0.9594	0.9317	0.8933	0.8432
25	0.9990	0.9974	0.9938	0.9869	0.9748	0.9554	0.9269	0.8878
26	0.9995	0.9987	0.9967	0.9925	0.9848	0.9718	0.9514	0.9221
27	0.9998	0.9994	0.9983	0.9959	0.9912	0.9827	0.9687	0.9475
28	0.9999	0.9997	0.9991	0.9978	0.9950	0.9897	0.9805	0.9657
29	1−	0.9999	0.9996	0.9989	0.9973	0.9941	0.9882	0.9782
30	1−	0.9999	0.9998	0.9994	0.9986	0.9967	0.9930	0.9865
31	1−	1−	0.9999	0.9997	0.9993	0.9982	0.9960	0.9919
32	1−	1−	1−	0.9999	0.9996	0.9990	0.9978	0.9953
33	1−	1−	1−	0.9999	0.9998	0.9995	0.9988	0.9973
34	1−	1−	1−	1−	0.9999	0.9998	0.9994	0.9985
35	1−	1−	1−	1−	1−	0.9999	0.9997	0.9992
36	1−	1−	1−	1−	1−	0.9999	0.9998	0.9996
37	1−	1−	1−	1−	1−	1−	0.9999	0.9998
38	1−	1−	1−	1−	1−	1−	1−	0.9999
39	1−	1−	1−	1−	1−	1−	1−	0.9999
40	1−	1−	1−	1−	1−	1−	1−	1−

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). For P(X ≥ x) use 1 − P(X ≤ x − 1).
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 at most 4 events next month is row 4, column λ = 2.5: 0.8912. The chance of 5 or more is 1 − 0.8912 = 0.1088, which could serve as a threshold for flagging an unusual month.

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

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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! · P(X ≤ x) = Σ_k=0^x P(X = k)

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.

Cumulative Poisson Distribution Table: P(X ≤ x)

Cumulative 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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