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Beta Function — Definition, Formula & Examples

The Beta function is a special function of two positive variables that arises frequently in calculus, probability, and statistics. It is defined by a specific definite integral and is closely related to the Gamma function.

For positive real numbers pp and qq, the Beta function is defined as B(p,q)=∫01t p−1(1−t) q−1 dtB(p,q) = \int_0^1 t^{\,p-1}(1-t)^{\,q-1}\,dt. It satisfies the identity B(p,q)=Γ(p) Γ(q)Γ(p+q)B(p,q) = \frac{\Gamma(p)\,\Gamma(q)}{\Gamma(p+q)}, where Γ\Gamma denotes the Gamma function.

Key Formula

B(p,q)=∫01t p−1(1−t) q−1 dt=Γ(p) Γ(q)Γ(p+q)B(p,q) = \int_0^1 t^{\,p-1}(1-t)^{\,q-1}\,dt = \frac{\Gamma(p)\,\Gamma(q)}{\Gamma(p+q)}
Where:
  • pp = First positive real parameter
  • qq = Second positive real parameter
  • Γ\Gamma = The Gamma function, which generalizes the factorial

How It Works

To evaluate the Beta function at specific values, you can either compute the integral directly or use the Gamma function identity. For positive integers, Γ(n)=(n−1)!\Gamma(n) = (n-1)!, which makes computation straightforward. The Beta function is symmetric: B(p,q)=B(q,p)B(p,q) = B(q,p), a fact you can verify by substituting u=1−tu = 1-t in the integral. This symmetry and the Gamma connection make it a powerful tool for evaluating integrals that would otherwise be difficult to handle.

Worked Example

Problem: Evaluate B(3, 4).
Recall the Gamma identity: Use the relationship between the Beta and Gamma functions.
B(3,4)=Γ(3) Γ(4)Γ(7)B(3,4) = \frac{\Gamma(3)\,\Gamma(4)}{\Gamma(7)}
Evaluate each Gamma value: Since p and q are positive integers, use the fact that Γ(n) = (n−1)!.
Γ(3)=2!=2,Γ(4)=3!=6,Γ(7)=6!=720\Gamma(3) = 2! = 2,\quad \Gamma(4) = 3! = 6,\quad \Gamma(7) = 6! = 720
Compute the result: Substitute and simplify.
B(3,4)=2⋅6720=12720=160B(3,4) = \frac{2 \cdot 6}{720} = \frac{12}{720} = \frac{1}{60}
Answer: B(3,4)=160B(3,4) = \dfrac{1}{60}

Why It Matters

The Beta function is the foundation of the Beta probability distribution, which is widely used in Bayesian statistics and machine learning to model proportions and probabilities. In advanced calculus courses, it provides an elegant shortcut for evaluating integrals of the form ∫01ta(1−t)b dt\int_0^1 t^a(1-t)^b\,dt without repeated integration by parts.

Common Mistakes

Mistake: Forgetting to subtract 1 from the exponents when matching an integral to the Beta function.
Correction: The integrand is tp−1(1−t)q−1t^{p-1}(1-t)^{q-1}, so if your integral has t5(1−t)3t^5(1-t)^3, then p=6p = 6 and q=4q = 4, not p=5p = 5 and q=3q = 3.