📘 Beta Distribution

🧭 When to Use the Beta

• Modeling a **proportion / probability** in \([0,1]\), e.g., pass rate, click-through rate, yield, defect fraction.
• Modeling a **bounded quantity** on \([a,b]\) via a scaled Beta (linear transform of \([0,1]\)).
• As a **Bayesian prior/posterior** for binomial proportion \(p\) (conjugate prior).
• Flexible shapes controlled by \(\alpha\) and \(\beta\): U-shaped, uniform, unimodal, skewed.

📐 Definitions & Key Formulas

Standard Beta (on [0,1]): \(X \sim \mathrm{Beta}(\alpha,\beta)\) with \(\alpha,\beta>0\)

• PDF: \[ f(x)=\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \quad 0<x<1 \] where \(B(\alpha,\beta)=\dfrac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}\).
• CDF: \(F(x)=I_x(\alpha,\beta)\) (regularized incomplete beta).
• Mean: \(\mathbb{E}[X]=\dfrac{\alpha}{\alpha+\beta}\)
• Variance: \(\mathrm{Var}(X)=\dfrac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\)
• Mode (if \(\alpha,\beta>1\)): \(\dfrac{\alpha-1}{\alpha+\beta-2}\)

Scaled Beta (on [a,b]): if \(Y\in[a,b]\), set \(X=\dfrac{Y-a}{b-a}\in[0,1]\). Then \(Y\sim \mathrm{ScaledBeta}(\alpha,\beta;a,b)\) and

• \(f_Y(y)=\dfrac{1}{b-a}\, f_X\!\left(\dfrac{y-a}{b-a}\right)\), \(\quad a<y<b\)
• \(\mathbb{E}[Y]=a+(b-a)\dfrac{\alpha}{\alpha+\beta}\)
• \(\mathrm{Var}(Y)=(b-a)^2 \dfrac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\)
• Quantile: \(y_p = a+(b-a)\,x_p\) where \(x_p\) is the Beta quantile in \([0,1]\).

Excel mapping: BETA.DIST(x, alpha, beta, cumulative, [A], [B]) and BETA.INV(p, alpha, beta, [A], [B]). Omit [A], [B] for the standard Beta on \([0,1]\). Include them for a scaled Beta on \([A,B]\).

🛠️ Worked Example 1 — Probability with Standard Beta

Scenario. A course’s pass rate \(P\) is modeled as \(P\sim \mathrm{Beta}(\alpha=8,\ \beta=4)\). What is \(P(P\ge 0.7)\)? Also report the mean and SD.

Step 1 — Mean/SD (sanity check)

• Mean \(=\alpha/(\alpha+\beta)=8/12=0.6667\).
• Variance \(=\alpha\beta/[(\alpha+\beta)^2(\alpha+\beta+1)] = 8\cdot4/(12^2\cdot 13)=32/1872=0.017094\). Hence SD \(\approx \sqrt{0.017094}=0.1308\).

Step 2 — Tail probability

\[ P(P\ge 0.7)=1-F(0.7)=1-I_{0.7}(8,4). \] (Use Excel to evaluate.)

Excel

=1 - BETA.DIST(0.7, 8, 4, TRUE)

Answer: \(P(P\ge 0.7)\approx\) (Excel will give the numeric value)

Interpretation: With mean ~0.667 and SD ~0.131, a 0.7 pass rate is a modest right-tail event.

🧰 Worked Example 2 — Warranty Cutoff with Scaled Beta

Scenario. A device’s **efficiency** \(Y\) lies in \([a,b]=[60\%,100\%]\). Suppose the rescaled proportion \(X=\dfrac{Y-0.60}{0.40}\) follows \(X\sim\mathrm{Beta}(\alpha=3,\ \beta=7)\). Find the **10th percentile** \(y_{0.10}\) to set a “bottom-10%” warranty threshold.

Step 1 — Use the scaled quantile transform

\[ y_p = a + (b-a)\,x_p, \quad x_p = \mathrm{BetaInv}(p;\alpha,\beta). \] With \(a=0.60,\ b=1.00,\ (b-a)=0.40\).

Step 2 — Compute \(x_{0.10}\) in \([0,1]\)

=BETA.INV(0.10, 3, 7)

Excel returns \(x_{0.10}\) (a number around 0.17–0.18 for these parameters).

Step 3 — Scale back to \([0.60,1.00]\)

\[ y_{0.10}=0.60 + 0.40 \times x_{0.10}. \]

=0.60 + 0.40 * BETA.INV(0.10, 3, 7)

Answer: \(y_{0.10}\) ≈ 0.60 + 0.40×(Excel value)

Interpretation: Only 10% of units fall below this efficiency. Use it as a conservative warranty cutoff.

🧾 Excel Quick Reference (Beta)

• Standard Beta (on [0,1])
• CDF: =BETA.DIST(x, alpha, beta, TRUE)
• PDF: =BETA.DIST(x, alpha, beta, FALSE)
• Quantile: =BETA.INV(p, alpha, beta)
• Mean: =alpha / (alpha + beta)
• Variance: =alpha*beta / ((alpha+beta)^2 * (alpha+beta+1))
• Scaled Beta (on [A,B])
• CDF \(P(Y\le y)\): =BETA.DIST( (y-A)/(B-A), alpha, beta, TRUE )
• PDF: =BETA.DIST( (y-A)/(B-A), alpha, beta, FALSE ) / (B-A)
• Quantile \(y_p\): =A + (B-A) * BETA.INV(p, alpha, beta)
• Mean: =A + (B-A) * alpha/(alpha+beta)
• Variance: =(B-A)^2 * alpha*beta / ((alpha+beta)^2 * (alpha+beta+1))

🧪 Beta Explorer (try it yourself)