How to read the table: each cell shows P(X ≤ x) for the selected parameters α and β. The row gives the first decimal digit of x; the column adds the second (×0.01). The range is x ∈ [0.00, 0.99].
Click a cell to read the value.
P(X ≤ x) for x from 0.00 to 0.99
| x | .00 | .01 | .02 | .03 | .04 | .05 | .06 | .07 | .08 | .09 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0.0 | — | 0.0003 | 0.0012 | 0.0026 | 0.0047 | 0.0073 | 0.0104 | 0.0140 | 0.0182 | 0.0228 |
| 0.1 | 0.0280 | 0.0336 | 0.0397 | 0.0463 | 0.0533 | 0.0608 | 0.0686 | 0.0769 | 0.0855 | 0.0946 |
| 0.2 | 0.1040 | 0.1138 | 0.1239 | 0.1344 | 0.1452 | 0.1563 | 0.1676 | 0.1793 | 0.1913 | 0.2035 |
| 0.3 | 0.2160 | 0.2287 | 0.2417 | 0.2548 | 0.2682 | 0.2818 | 0.2955 | 0.3094 | 0.3235 | 0.3377 |
| 0.4 | 0.3520 | 0.3665 | 0.3810 | 0.3957 | 0.4104 | 0.4253 | 0.4401 | 0.4551 | 0.4700 | 0.4850 |
| 0.5 | 0.5000 | 0.5150 | 0.5300 | 0.5449 | 0.5599 | 0.5747 | 0.5896 | 0.6043 | 0.6190 | 0.6335 |
| 0.6 | 0.6480 | 0.6623 | 0.6765 | 0.6906 | 0.7045 | 0.7182 | 0.7318 | 0.7452 | 0.7583 | 0.7713 |
| 0.7 | 0.7840 | 0.7965 | 0.8087 | 0.8207 | 0.8324 | 0.8438 | 0.8548 | 0.8656 | 0.8761 | 0.8862 |
| 0.8 | 0.8960 | 0.9054 | 0.9145 | 0.9231 | 0.9314 | 0.9393 | 0.9467 | 0.9537 | 0.9603 | 0.9664 |
| 0.9 | 0.9720 | 0.9772 | 0.9818 | 0.9860 | 0.9896 | 0.9928 | 0.9953 | 0.9974 | 0.9988 | 0.9997 |
How to read the table: each cell shows f(x) = x^(α−1) · (1−x)^(β−1) / B(α,β). For α < 1 or β < 1 the density may diverge at the endpoints (very large values are shown as "∞").
Click a cell to read the value.
f(x) for x from 0.00 to 0.99
| x | .00 | .01 | .02 | .03 | .04 | .05 | .06 | .07 | .08 | .09 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0.0 | — | 0.0594 | 0.1176 | 0.1746 | 0.2304 | 0.2850 | 0.3384 | 0.3906 | 0.4416 | 0.4914 |
| 0.1 | 0.5400 | 0.5874 | 0.6336 | 0.6786 | 0.7224 | 0.7650 | 0.8064 | 0.8466 | 0.8856 | 0.9234 |
| 0.2 | 0.9600 | 0.9954 | 1.0296 | 1.0626 | 1.0944 | 1.1250 | 1.1544 | 1.1826 | 1.2096 | 1.2354 |
| 0.3 | 1.2600 | 1.2834 | 1.3056 | 1.3266 | 1.3464 | 1.3650 | 1.3824 | 1.3986 | 1.4136 | 1.4274 |
| 0.4 | 1.4400 | 1.4514 | 1.4616 | 1.4706 | 1.4784 | 1.4850 | 1.4904 | 1.4946 | 1.4976 | 1.4994 |
| 0.5 | 1.5000 | 1.4994 | 1.4976 | 1.4946 | 1.4904 | 1.4850 | 1.4784 | 1.4706 | 1.4616 | 1.4514 |
| 0.6 | 1.4400 | 1.4274 | 1.4136 | 1.3986 | 1.3824 | 1.3650 | 1.3464 | 1.3266 | 1.3056 | 1.2834 |
| 0.7 | 1.2600 | 1.2354 | 1.2096 | 1.1826 | 1.1544 | 1.1250 | 1.0944 | 1.0626 | 1.0296 | 0.9954 |
| 0.8 | 0.9600 | 0.9234 | 0.8856 | 0.8466 | 0.8064 | 0.7650 | 0.7224 | 0.6786 | 0.6336 | 0.5874 |
| 0.9 | 0.5400 | 0.4914 | 0.4416 | 0.3906 | 0.3384 | 0.2850 | 0.2304 | 0.1746 | 0.1176 | 0.0594 |
How to read the table: each cell shows the quantile x such that P(X ≤ x) = p. Rows: different values of α. Columns: cumulative probabilities p. Select β with the selector.
Click a cell to read the value.
Quantiles Be⁻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 | 0.0006 | 0.0025 | 0.0100 | 0.0625 | 0.2500 | 0.5625 | 0.8100 | 0.9025 | 0.9506 | 0.9801 |
| 1 | 0.0100 | 0.0250 | 0.0500 | 0.1000 | 0.2500 | 0.5000 | 0.7500 | 0.9000 | 0.9500 | 0.9750 | 0.9900 |
| 1.5 | 0.0464 | 0.0855 | 0.1357 | 0.2154 | 0.3969 | 0.6300 | 0.8255 | 0.9322 | 0.9664 | 0.9833 | 0.9933 |
| 2 | 0.1000 | 0.1581 | 0.2236 | 0.3162 | 0.5000 | 0.7071 | 0.8660 | 0.9487 | 0.9747 | 0.9874 | 0.9950 |
| 2.5 | 0.1585 | 0.2287 | 0.3017 | 0.3981 | 0.5743 | 0.7579 | 0.8913 | 0.9587 | 0.9797 | 0.9899 | 0.9960 |
| 3 | 0.2154 | 0.2924 | 0.3684 | 0.4642 | 0.6300 | 0.7937 | 0.9086 | 0.9655 | 0.9830 | 0.9916 | 0.9967 |
| 4 | 0.3162 | 0.3976 | 0.4729 | 0.5623 | 0.7071 | 0.8409 | 0.9306 | 0.9740 | 0.9873 | 0.9937 | 0.9975 |
| 5 | 0.3981 | 0.4782 | 0.5493 | 0.6310 | 0.7579 | 0.8706 | 0.9441 | 0.9791 | 0.9898 | 0.9949 | 0.9980 |
| 7 | 0.5179 | 0.5904 | 0.6518 | 0.7197 | 0.8203 | 0.9057 | 0.9597 | 0.9851 | 0.9927 | 0.9964 | 0.9986 |
| 10 | 0.6310 | 0.6915 | 0.7411 | 0.7943 | 0.8706 | 0.9330 | 0.9716 | 0.9895 | 0.9949 | 0.9975 | 0.9990 |
How to use these tables
Beta Distribution
The Beta distribution Be(α, β) has support on [0, 1] and is very flexible for modeling proportions and probabilities. Its density function is:
\( f(x;\alpha,\beta) = \dfrac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \quad x \in (0,1) \)
where B(α, β) = Γ(α)Γ(β)/Γ(α+β) is the beta function. The mean is μ = α/(α+β) and the variance σ² = αβ / [(α+β)²(α+β+1)].
Shapes of the distribution
- α = β = 1: Uniform(0,1) distribution.
- α = β > 1: unimodal and symmetric around 0.5.
- α > β: left-skewed (mean > 0.5).
- α < β: right-skewed (mean < 0.5).
- α < 1 or β < 1: U-shaped (bimodal at the endpoints).
Relationship with F and t
If F ~ F(2α, 2β) then X = αF/(β + αF) ~ Be(α, β). The regularized incomplete beta function is also the CDF of the F distribution and of Student's t distribution.