Gamma Distribution Table

How to read the table: each cell shows P(X ≤ x) for the selected shape parameter α and scale parameter β. The row gives the integer part of x; the column adds the first decimal (×0.1).

α (shape): β (scale):

Click a cell to read the value.

P(X ≤ x) for x from 0.0 to 30.9

x	.0	.1	.2	.3	.4	.5	.6	.7	.8	.9
0.0	0.0000	0.0047	0.0175	0.0369	0.0616	0.0902	0.1219	0.1558	0.1912	0.2275
1.0	0.2642	0.3010	0.3374	0.3732	0.4082	0.4422	0.4751	0.5068	0.5372	0.5663
2.0	0.5940	0.6204	0.6454	0.6691	0.6916	0.7127	0.7326	0.7513	0.7689	0.7854
3.0	0.8009	0.8153	0.8288	0.8414	0.8532	0.8641	0.8743	0.8838	0.8926	0.9008
4.0	0.9084	0.9155	0.9220	0.9281	0.9337	0.9389	0.9437	0.9482	0.9523	0.9561
5.0	0.9596	0.9628	0.9658	0.9686	0.9711	0.9734	0.9756	0.9776	0.9794	0.9811
6.0	0.9826	0.9841	0.9854	0.9866	0.9877	0.9887	0.9897	0.9905	0.9913	0.9920
7.0	0.9927	0.9933	0.9939	0.9944	0.9949	0.9953	0.9957	0.9961	0.9964	0.9967
8.0	0.9970	0.9972	0.9975	0.9977	0.9979	0.9981	0.9982	0.9984	0.9985	0.9986
9.0	0.9988	0.9989	0.9990	0.9991	0.9991	0.9992	0.9993	0.9993	0.9994	0.9995
10.0	0.9995	0.9995	0.9996	0.9996	0.9997	0.9997	0.9997	0.9997	0.9998	0.9998
11.0	0.9998	0.9998	0.9998	0.9998	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999
12.0	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	1.0000	1.0000	1.0000	1.0000
13.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
14.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
15.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
16.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
17.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
18.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
19.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
20.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
21.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
22.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
23.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
24.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
25.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
26.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
27.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
28.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
29.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
30.0	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

How to read the table: each cell shows f(x) = x^(α−1) · e^(−x/β) / (β^α · Γ(α)) for the selected parameters. The density is 0 for x ≤ 0.

α (shape): β (scale):

Click a cell to read the value.

f(x) for x from 0.0 to 30.9

x	.0	.1	.2	.3	.4	.5	.6	.7	.8	.9
0.0	0.0000	0.0905	0.1637	0.2222	0.2681	0.3033	0.3293	0.3476	0.3595	0.3659
1.0	0.3679	0.3662	0.3614	0.3543	0.3452	0.3347	0.3230	0.3106	0.2975	0.2842
2.0	0.2707	0.2572	0.2438	0.2306	0.2177	0.2052	0.1931	0.1815	0.1703	0.1596
3.0	0.1494	0.1397	0.1304	0.1217	0.1135	0.1057	0.0984	0.0915	0.0850	0.0789
4.0	0.0733	0.0679	0.0630	0.0583	0.0540	0.0500	0.0462	0.0427	0.0395	0.0365
5.0	0.0337	0.0311	0.0287	0.0265	0.0244	0.0225	0.0207	0.0191	0.0176	0.0162
6.0	0.0149	0.0137	0.0126	0.0116	0.0106	0.0098	0.0090	0.0082	0.0076	0.0070
7.0	0.0064	0.0059	0.0054	0.0049	0.0045	0.0041	0.0038	0.0035	0.0032	0.0029
8.0	0.0027	0.0025	0.0023	0.0021	0.0019	0.0017	0.0016	0.0014	0.0013	0.0012
9.0	0.0011	0.0010	0.0009	0.0009	0.0008	0.0007	0.0007	0.0006	0.0005	0.0005
10.0	0.0005	0.0004	0.0004	0.0003	0.0003	0.0003	0.0003	0.0002	0.0002	0.0002
11.0	0.0002	0.0002	0.0002	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001
12.0	0.0001	0.0001	0.0001	0.0001	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000
13.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
14.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
15.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
16.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
17.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
18.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
19.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
20.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
21.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
22.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
23.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
24.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
25.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
26.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
27.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
28.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
29.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
30.0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

How to read the table: each cell shows the quantile x such that P(X ≤ x) = p. Rows: different values of α (shape). Columns: cumulative probabilities p. Select β (scale) with the selector.

β (scale):

Click a cell to read the value.

Quantiles Γ⁻¹(p; α, β) — P(X ≤ x) = p

α \ p	0.01	0.025	0.05	0.1	0.25	0.5	0.75	0.9	0.95	0.975	0.99
0.5	0.0001	0.0005	0.0020	0.0079	0.0508	0.2275	0.6617	1.3528	1.9207	2.5119	3.3174
1	0.0101	0.0253	0.0513	0.1054	0.2877	0.6931	1.3863	2.3026	2.9957	3.6889	4.6052
1.5	0.0574	0.1079	0.1759	0.2922	0.6063	1.1830	2.0542	3.1257	3.9074	4.6742	5.6724
2	0.1486	0.2422	0.3554	0.5318	0.9613	1.6783	2.6926	3.8897	4.7439	5.5716	6.6384
2.5	0.2771	0.4156	0.5727	0.8052	1.3373	2.1757	3.3128	4.6182	5.5352	6.4163	7.5431
3	0.4360	0.6187	0.8177	1.1021	1.7273	2.6741	3.9204	5.3223	6.2958	7.2247	8.4059
4	0.8232	1.0899	1.3663	1.7448	2.5353	3.6721	5.1094	6.6808	7.7537	8.7673	10.0451
5	1.2791	1.6235	1.9701	2.4326	3.3686	4.6709	6.2744	7.9936	9.1535	10.2416	11.6046
6	1.7853	2.2019	2.6130	3.1519	4.2192	5.6702	7.4227	9.2747	10.5130	11.6683	13.1085
7	2.3302	2.8144	3.2853	3.8948	5.0827	6.6696	8.5585	10.5321	11.8424	13.0595	14.5706
8	2.9061	3.4538	3.9808	4.6561	5.9561	7.6692	9.6844	11.7709	13.1481	14.4227	16.0000
9	3.5075	4.1154	4.6952	5.4325	6.8376	8.6690	10.8024	12.9947	14.4346	15.7632	17.4027
10	4.1302	4.7954	5.4254	6.2213	7.7259	9.6687	11.9138	14.2060	15.7052	17.0848	18.7831
15	7.4767	8.3954	9.2463	10.2996	12.2388	14.6680	17.3999	20.1280	21.8865	23.4896	25.4461
20	11.0821	12.2165	13.2547	14.5253	16.8301	19.6677	22.8080	25.9025	27.8792	29.6709	31.8454

How to use these tables

Gamma Distribution

The Gamma distribution Γ(α, β) generalizes the Exponential distribution. Its density function is:

\( f(x;\alpha,\beta) = \dfrac{x^{\alpha-1}\,e^{-x/\beta}}{\beta^\alpha\,\Gamma(\alpha)}, \quad x > 0, \;\alpha > 0,\; \beta > 0 \)

The mean is μ = αβ and the variance σ² = αβ². The parameter α controls the shape and β the scale.

Special cases

Γ(1, β): coincides with the Exponential distribution with mean β.
Γ(ν/2, 2): coincides with the Chi-square distribution with ν degrees of freedom.
Γ(k, 1) with integer k: sum of k independent Exponentials with mean 1 (Erlang distribution).

Parameterization

This table uses the (α, β) parameterization where β is the scale and 1/β is the rate. Check the parameterization used by your software before comparing values.

How to use these tables

Gamma Distribution

Special cases

Parameterization

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