Beta Distribution

Why This Matters

The Beta distribution is the parametric family of densities on the unit interval. Two reasons it earns its own page rather than a single line in a survey:

1. It is the conjugate prior for any likelihood of the form "$k$ successes in $n$ trials". Bernoulli, Binomial, and Negative Binomial all admit a Beta posterior. The update is among the cleanest in Bayesian statistics: add the number of successes to one shape and the number of failures to the other.
2. The order statistics of an i.i.d. sample from $\operatorname{Unif}(0,1)$ are Beta distributed. The $k$-th smallest value out of $n$ uniforms is $\operatorname{Beta}(k, n-k+1)$. This geometric construction predates the Bayesian interpretation by several decades and is the cleanest way to see why the density has its specific shape.

The Beta is also the marginal of any pair of components of a Dirichlet random vector. The Dirichlet generalizes Beta to the simplex of probability vectors; Beta is the two-category special case.

Definition

Definition

Beta Distribution$X\sim \mathrm{Beta}(\alpha ,\beta )$

A random variable $X$ has a Beta distribution with shape parameters $\alpha > 0$ and $\beta > 0$ if its density is

$f_X(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)},\qquad 0 < x < 1,$

where $B(\alpha,\beta) = \Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha+\beta)$ is the Beta function.

The density is supported on the open unit interval. The two shape parameters control the location and concentration of the mass: large $\alpha$ relative to $\beta$ pulls the mass toward $1$; large $\beta$ relative to $\alpha$ pulls it toward $0$; large $\alpha + \beta$ concentrates the mass.

The case $\alpha = \beta = 1$ is the Uniform(0,1) distribution. The case $\alpha = \beta < 1$ is a U-shaped density with mass concentrated at both endpoints; the case $\alpha = \beta > 1$ is symmetric and unimodal at $1/2$. Asymmetric shape parameters give skewed densities, with the mode at $(\alpha-1)/(\alpha+\beta-2)$ when both shapes exceed one.

The easiest way to read the parameters is to separate direction from concentration:

The ratio $\alpha/(\alpha+\beta)$ sets the center. The sum $\alpha+\beta$ sets how hard the density resists movement away from that center. This is why $\operatorname{Beta}(2,2)$ and $\operatorname{Beta}(200,200)$ have the same mean but behave differently in a Bayesian update: the first moves after a few observations; the second needs a large sample before the posterior mean changes much.

Density and Moments

Proposition

Beta Mean and Variance

Statement

For $X\sim\operatorname{Beta}(\alpha,\beta)$, $\mathbb{E}[X] = \frac{\alpha}{\alpha+\beta},\qquad \operatorname{Var}(X) = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}.$ More generally, $\mathbb{E}[X^k] = B(\alpha+k,\beta)/B(\alpha,\beta)$ for every positive integer $k$.

Intuition

The mean depends only on the ratio of shapes, but the variance shrinks as $\alpha + \beta$ grows. Two Beta densities with the same mean can have very different variances; the sum $\alpha + \beta$ is the concentration parameter.

Proof Sketch

Direct computation: $\mathbb{E}[X^k] = \int_0^1 x^k\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}\,dx = \frac{B(\alpha+k,\beta)}{B(\alpha,\beta)} = \frac{\Gamma(\alpha+k)\Gamma(\alpha+\beta)}{\Gamma(\alpha+\beta+k)\Gamma(\alpha)}.$ For $k = 1$, $\Gamma(\alpha+1)/\Gamma(\alpha) = \alpha$ and $\Gamma(\alpha+\beta)/\Gamma(\alpha+\beta+1) = 1/(\alpha+\beta)$, giving the mean. The variance follows by computing $\mathbb{E}[X^2]$ and subtracting the squared mean.

Why It Matters

The two parameters together control "where the mass is" (through the mean) and "how concentrated it is" (through $\alpha+\beta$). In Bayesian inference, $\alpha + \beta$ acts as a "pseudo-count" of prior observations; the larger it is, the harder it is for new data to move the posterior.

Failure Mode

The moment formulas require $\alpha > 0$ and $\beta > 0$. If either shape is nonpositive, $B(\alpha,\beta)$ is not the normalizing constant of a probability density on $(0,1)$. For $\alpha < 1$ or $\beta < 1$, the density is unbounded at $0$ or $1$, although it remains integrable. Numerical evaluation near the boundaries requires log-density arithmetic; direct evaluation of $x^{\alpha-1}(1-x)^{\beta-1}$ can overflow near a singular endpoint and underflow when both shapes are large.

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Worked Examples

Example

Same mean, different concentration

Compare $X\sim\operatorname{Beta}(2,2)$ and $Y\sim\operatorname{Beta}(20,20)$. Both have mean $1/2$. The variances differ:

$$\operatorname{Var}(X)=\frac{2\cdot2}{4^2\cdot5}=0.05,\qquad \operatorname{Var}(Y)=\frac{20\cdot20}{40^2\cdot41}\approx0.0061.$$

The mechanism is the denominator $\alpha+\beta+1$. Multiplying both shapes by ten leaves the mean unchanged but increases the concentration. In a Beta-Bernoulli model, $\operatorname{Beta}(2,2)$ behaves like a weak prior centered at $1/2$; $\operatorname{Beta}(20,20)$ behaves like a prior with much more resistance. They encode the same location, not the same information.

Example

Posterior mean as a weighted average

Start with $p\sim\operatorname{Beta}(8,2)$, whose prior mean is $0.8$. Observe $S=6$ successes in $n=20$ Bernoulli trials. The posterior is $p\mid X\sim\operatorname{Beta}(8+6,\ 2+14)=\operatorname{Beta}(14,16).$ The posterior mean is $14/30\approx0.467$.

The same number appears from the weighted-average form:

$$\frac{\alpha_0+\beta_0}{\alpha_0+\beta_0+n}\cdot \frac{\alpha_0}{\alpha_0+\beta_0} + \frac{n}{\alpha_0+\beta_0+n}\cdot\frac{S}{n} = \frac{10}{30}\cdot0.8+\frac{20}{30}\cdot0.3 =0.467.$$

The posterior does not average the prior density with the empirical frequency. It averages two probability estimates with weights given by prior concentration and sample size.

Example

Uniform order statistic by counting gaps

Let $U_1,\dots,U_5$ be i.i.d. uniforms and consider the second-smallest value $U_{(2)}$. The theorem gives $U_{(2)}\sim\operatorname{Beta}(2,4)$, so the mean is $2/6=1/3$.

The density can be read without memorizing the formula. For $U_{(2)}$ to land near $x$, one observation must fall below $x$, one must land near $x$, and three must fall above $x$. The number of assignments is $5!/(1!1!3!)=20$, so $f_{U_{(2)}}(x)=20x(1-x)^3,\qquad0<x<1.$ This is exactly the $\operatorname{Beta}(2,4)$ density. The exponent of $x$ counts observations forced below the statistic; the exponent of $1-x$ counts observations forced above it.

Beta as a Uniform Order Statistic

Theorem

Order Statistics of Uniforms Are Beta

Statement

Let $U_1,\dots,U_n$ be i.i.d. $\operatorname{Unif}(0,1)$ and let $U_{(k)}$ denote the $k$-th smallest value. Then $U_{(k)}\sim\operatorname{Beta}(k,\,n-k+1).$

Intuition

For $U_{(k)}$ to lie in a small interval near $x$, we need $k-1$ uniforms to fall below it, the order statistic itself to fall near $x$, and $n-k$ uniforms to fall above it. The density at $x$ is $\binom{n}{k-1,1,n-k}x^{k-1}(1-x)^{n-k}$ times the density of a single uniform near $x$, which simplifies to the Beta density with $\alpha = k$ and $\beta = n - k + 1$.

Proof Sketch

The joint density of $U_{(1)},\dots,U_{(n)}$ is $n!\cdot\mathbf{1}\{0 \le u_1 \le\cdots\le u_n\le 1\}$. To compute the marginal density of $U_{(k)}$, fix $u_{(k)} = x$ and integrate over the other coordinates. The lower $k-1$ uniforms lie in $[0, x]$ (volume $x^{k-1}/(k-1)!$), and the upper $n-k$ lie in $[x, 1]$ (volume $(1-x)^{n-k}/(n-k)!$). Combining: $f_{U_{(k)}}(x) = n!\cdot\frac{x^{k-1}}{(k-1)!}\cdot\frac{(1-x)^{n-k}}{(n-k)!} = \frac{n!}{(k-1)!(n-k)!}x^{k-1}(1-x)^{n-k}.$ The constant $n!/[(k-1)!(n-k)!] = 1/B(k, n-k+1)$ by the relationship between the Beta function and binomial coefficients. So $U_{(k)}\sim\operatorname{Beta}(k,n-k+1)$.

Why It Matters

This gives a purely geometric origin for the Beta family that does not depend on Bayesian thinking. The Beta is the natural distribution of "the position of the $k$-th of $n$ uniformly scattered points on the unit interval". The conjugate-prior role for the Bernoulli falls out from the same combinatorial structure: the posterior of $p$ given $k$ successes in $n$ trials is the predictive distribution of the next ranked position.

Failure Mode

The result requires independent uniforms with the same distribution. If the sample comes from a non-uniform continuous distribution with CDF $F$, then $F(X_{(k)})$ is Beta, but $X_{(k)}$ itself is not. If the observations are dependent, the multinomial counting argument fails because the counts below and above $x$ are no longer binomial. Ties also change the statement: for discrete samples there may be no unique $k$-th location with a density on $(0,1)$.

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Conjugate Prior for the Bernoulli and Binomial

Theorem

Beta-Bernoulli Conjugacy

Statement

Let $X_1,\dots,X_n\sim\operatorname{Bern}(p)$ be i.i.d., let $S = \sum X_i$ be the total successes, and let the prior be $p\sim\operatorname{Beta}(\alpha_0,\beta_0)$. Then $p\mid X_1,\dots,X_n\sim\operatorname{Beta}(\alpha_0 + S,\ \beta_0 + n - S).$ For an observation of $S\sim\operatorname{Bin}(n,p)$ as a single sufficient summary, the same posterior holds.

Intuition

A Beta prior contributes $\alpha_0 - 1$ pseudo-successes and $\beta_0 - 1$ pseudo-failures. Real data adds real successes and real failures. The posterior is Beta with shape parameters equal to total successes plus pseudo-successes plus one (and the same for failures).

Proof Sketch

The Bernoulli likelihood is $L(p) = \prod_{i=1}^n p^{X_i}(1-p)^{1-X_i} = p^S(1-p)^{n-S}.$ The Beta prior density is proportional to $p^{\alpha_0-1}(1-p)^{\beta_0-1}$. The posterior is proportional to the product: $\pi(p\mid X)\propto p^{\alpha_0+S-1}(1-p)^{\beta_0+n-S-1},$ which is the kernel of $\operatorname{Beta}(\alpha_0+S, \beta_0+n-S)$.

Why It Matters

Posterior mean is $(\alpha_0+S)/(\alpha_0+\beta_0+n)$, a precision-weighted blend of the prior mean $\alpha_0/(\alpha_0+\beta_0)$ and the MLE $S/n$. The blend weight on the MLE is $n/(\alpha_0+\beta_0+n)$, which approaches one as data accumulates. The Jeffreys prior for the Bernoulli is $\operatorname{Beta}(1/2, 1/2)$; the uniform prior is $\operatorname{Beta}(1, 1)$; the Haldane prior is $\operatorname{Beta}(0, 0)$ (improper). All three are admissible starting points with different bias-variance trade-offs.

Failure Mode

Conjugacy is a property of the likelihood-prior pair. With i.i.d. Bernoulli or Binomial sampling and a proper Beta prior, the posterior is Beta. If the trials are not conditionally independent given $p$, if each observation has its own probability $p_i$, or if the likelihood is a regression model such as logistic or probit regression, the Beta update no longer applies. The same failure occurs when the prior is improper without a proper posterior: for example, the Haldane prior with all successes or all failures leaves one posterior shape at zero.

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The conjugacy calculation is short because the Bernoulli likelihood and the Beta density use the same sufficient statistic. The data enter only through $S$ and $n-S$. This is not an accident: the Bernoulli family is a one-parameter exponential family, and the Beta prior is the conjugate prior obtained by matching the powers of $p$ and $1-p$. Once the likelihood has extra structure, such as covariates in logistic regression, the sufficient statistic is no longer just a pair of counts. The posterior cannot stay inside the two-shape Beta family because two numbers are not enough to store the information in the design matrix.

Example

Posterior predictive probability

Conjugacy also gives the next-trial predictive probability without numerical integration. If $p\mid X\sim\operatorname{Beta}(\alpha_0+S,\beta_0+n-S)$, then

$$P(X_{n+1}=1\mid X_1,\dots,X_n) =\mathbb{E}[p\mid X] =\frac{\alpha_0+S}{\alpha_0+\beta_0+n}.$$

For the earlier $S=6$, $n=20$, $\operatorname{Beta}(8,2)$ example, the predictive probability of success on the next trial is $14/30\approx0.467$. The plug-in MLE would use $6/20=0.3$; the prior-only prediction would use $8/10=0.8$. The posterior predictive value is between them because it averages over posterior uncertainty in $p$ instead of pretending that either the prior mean or the empirical frequency is exact.

For $m$ future Bernoulli trials, the predictive distribution of the future success count is Beta-Binomial, not Binomial with $p$ fixed at the posterior mean. The distinction is variance: integrating over $p$ adds uncertainty. If a model predicts the right average number of future successes but systematically understates the spread, the missing term is often this posterior uncertainty.

Connection to the Gamma Distribution

The Beta arises naturally from ratios of independent Gammas. If $X\sim\operatorname{Gamma}(\alpha,1)$ and $Y\sim\operatorname{Gamma}(\beta,1)$ are independent, then

$\frac{X}{X+Y}\sim\operatorname{Beta}(\alpha,\beta),$

and this ratio is independent of $X+Y\sim\operatorname{Gamma}(\alpha+\beta,1)$. This identity is the reason the Beta function $B(\alpha,\beta) = \Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha+\beta)$ has the form it does: it is the Jacobian-adjusted ratio of Gamma normalizing constants. See gamma distribution for the Gamma side.

The same identity gives a simple way to sample from the Beta: draw $X$ and $Y$ from independent Gammas (which are sums of independent Exponentials when shapes are integers) and compute the ratio.

Maximum Likelihood Estimation

The MLE of $(\alpha,\beta)$ from an i.i.d. Beta sample has no closed form. The log-likelihood is

$\ell(\alpha,\beta) = -n\log B(\alpha,\beta) + (\alpha-1)\sum\log X_i + (\beta-1)\sum\log(1-X_i),$

and the score equations are

$\psi(\alpha) - \psi(\alpha+\beta) = \overline{\log X},\qquad \psi(\beta) - \psi(\alpha+\beta) = \overline{\log(1-X)},$

where $\psi$ is the digamma function and the overlines denote sample averages. These must be solved numerically (e.g., by Newton's method, with a starting point from the method-of-moments estimator). The Fisher information matrix is

$I(\alpha,\beta) = \begin{pmatrix} \psi'(\alpha) - \psi'(\alpha+\beta) & -\psi'(\alpha+\beta) \\ -\psi'(\alpha+\beta) & \psi'(\beta) - \psi'(\alpha+\beta) \end{pmatrix}.$

In Bayesian workflows the MLE is rarely used; the posterior is typically the object of interest, and the Beta-Bernoulli conjugacy gives the posterior in closed form regardless of how the prior was chosen.

Method of Moments (Closed Form)

The method-of-moments estimator has a closed form. With $\bar X_n$ the sample mean and $\hat\sigma^2_n$ the sample variance,

$\hat\alpha_{\text{MoM}} = \bar X_n\!\left(\frac{\bar X_n(1-\bar X_n)}{\hat\sigma^2_n} - 1\right),\qquad \hat\beta_{\text{MoM}} = (1-\bar X_n)\!\left(\frac{\bar X_n(1-\bar X_n)}{\hat\sigma^2_n} - 1\right).$

The estimator is consistent and almost always reasonable as a starting point for MLE iteration. See method of moments for the general framework.

The method-of-moments formula also names a boundary condition. A Beta distribution must satisfy $\operatorname{Var}(X) < \mathbb{E}[X](1-\mathbb{E}[X])$. If the empirical variance violates that inequality, the plug-in expression $\bar X_n(1-\bar X_n)/\hat\sigma_n^2 - 1$ is nonpositive, so at least one shape estimate is invalid. This can happen with data rounded to endpoints, with a misspecified sample that is not supported on $[0,1]$, or with a mixture that is too dispersed for a single Beta component. The failure is diagnostic: the problem is not Newton iteration; the one-component Beta model is the wrong summary for the sample.

Diagnostics and Boundary Cases

The Beta family is flexible on $[0,1]$, but it is still a two-parameter model. The first diagnostic is the mean-variance relation. If the mean is $m$ and the concentration is $\tau=\alpha+\beta$, then

$$\alpha=m\tau,\qquad \beta=(1-m)\tau,\qquad \operatorname{Var}(X)=\frac{m(1-m)}{\tau+1}.$$

For a fixed mean, the largest possible Beta variance is approached as $\tau\downarrow0$ and equals $m(1-m)$ in the limit. Any sample variance above that level cannot come from a single proper Beta distribution with that sample mean. The usual method-of-moments formula is just this relation solved backward: $\tau=\frac{m(1-m)}{v}-1.$ A negative estimate of $\tau$ is not a numerical glitch; it says the observed spread is too large for the family.

Example

When a single Beta fit goes degenerate

Suppose a dataset on $[0,1]$ has sample mean $\bar x=0.5$ and sample variance $\hat v=0.24$. The method-of-moments concentration is $\tau=\frac{0.5(1-0.5)}{0.24}-1=\frac{0.25}{0.24}-1\approx0.0417,$ so $\alpha=\beta=\tau\bar x\approx0.021$. That is a deep U-shaped Beta with almost all of its mass pressed against $0$ and $1$. A fit this close to the ceiling $m(1-m)=0.25$ is a warning, not a model: the data is nearly as dispersed as a two-point mass, and a single smooth Beta density describes it poorly.

The ceiling is sharp, and it bounds the data itself, not just the Beta family. A reported sample variance above $0.25$ at mean $0.5$ would force $\tau<0$, i.e. negative shape parameters, which no Beta admits. For values genuinely confined to $[0,1]$ that case cannot arise: Popoviciu caps the variance at $(b-a)^2/4$ and Bhatia-Davis at $(b-m)(m-a)$, both equal to $0.25$ here. These ceilings bound the population ($1/n$) variance; the unbiased ($n-1$) estimator can exceed $0.25$ on data genuinely inside $[0,1]$, so read $\hat v$ as the $1/n$ form throughout. Seeing $\tau<0$ under that convention therefore means the values are not actually in $[0,1]$ or $\hat v$ was mis-estimated. When the sample piles up at exact zeros and ones, use a zero-one-inflated model, a mixture model, or a data-generating model for the counts that produced the proportions.

Endpoint observations need separate handling. The Beta density is defined on the open interval $(0,1)$, so $\log X_i$ and $\log(1-X_i)$ in the likelihood are finite only when every observation lies strictly between zero and one. Exact zeros and ones are common when the data are empirical rates: a small school may have zero failures in a year; a small experiment may have all successes. Those observations are not illegal data, but they do not fit the continuous Beta likelihood without a measurement model. The usual fixes are explicit: jittering is a preprocessing convention, not a likelihood; adding counts is a Binomial model; adding point masses at zero and one is a zero-one-inflated Beta model.

Example

Exact endpoints versus small neighborhoods

Take $p\sim\operatorname{Beta}(1/2,1/2)$. The density is $f(p)=\frac{1}{\pi\sqrt{p(1-p)}}.$ It diverges as $p\downarrow0$ and as $p\uparrow1$, but $P(p=0)=P(p=1)=0$. A simulation will produce many values very close to the endpoints and no exact endpoint unless the random-number generator rounds.

That distinction changes modeling decisions. If observed records contain literal zeros because a rate was computed as $0/7$, the underlying Bernoulli probability might still be inside $(0,1)$ and the count model is appropriate. If observed records contain structural zeros, such as a category that cannot occur for a subgroup, the Beta model is wrong because the atom at zero is part of the phenomenon.

The second diagnostic is whether the data-generating story is exchangeable. Beta-Bernoulli conjugacy assumes one latent $p$ for all trials. If the first ten observations come from one population and the next ten from another, a single posterior shape pair hides heterogeneity by averaging it away. In that case a hierarchical model is the natural extension: each group has its own $p_j$, and the group probabilities share a higher-level distribution. The ordinary Beta update is still useful locally, but it is no longer the whole model.

Choosing a Beta Prior

The practical prior-choice question is not "which Beta is neutral?" but "what mean and concentration are defensible before seeing these data?" The mean $m=\alpha/(\alpha+\beta)$ gives the center. The concentration $\tau=\alpha+\beta$ gives the prior sample-size scale. Once those are chosen, $\alpha=m\tau,\qquad \beta=(1-m)\tau.$

For example, suppose a baseline conversion rate is near $0.04$, but the new experiment is different enough that the prior should be weak. Taking $m=0.04$ and $\tau=25$ gives $\operatorname{Beta}(1,24)$. The prior mean is $0.04$, but after $n=500$ visitors it receives only $25/(25+500)$ of the posterior-mean weight. If the old experiments are closely matched to the new one, $\tau=500$ would encode a much stronger prior, giving $\operatorname{Beta}(20,480)$.

This parameterization also exposes bad defaults. A uniform $\operatorname{Beta}(1,1)$ prior has mean $0.5$. For rare-event problems, that center is far from plausible, though the concentration is weak. Jeffreys' $\operatorname{Beta}(1/2,1/2)$ prior is invariant under smooth reparameterization and puts more density near the endpoints. It is often a better default for pure Bernoulli inference, but it is not a substitute for domain knowledge when base rates are known.

Prior sensitivity should be reported on the scale readers care about. Do not only list $\alpha$ and $\beta$. Report the prior mean, concentration, posterior mean after the observed counts, and predictive probability for the next trial. If two plausible priors lead to the same decision after the data, the inference is insensitive to that prior choice. If they do not, the page should say that the data are not yet strong enough to dominate the prior.

Common Confusions

Watch Out

Pseudo-counts versus real counts

A Beta(\alpha_0, \beta_0) prior is sometimes described as "equivalent to having seen $\alpha_0 - 1$ successes and $\beta_0 - 1$ failures". This intuition is right in expectation but wrong in variance: the prior also has a "concentration" $\alpha_0 + \beta_0$ that real data cannot match unless the data sample is at least that large. Conjugate priors compress prior information into a sufficient statistic, but they do not replace it with imaginary observations.

Watch Out

Beta(1, 1) is the uniform distribution

The Beta(1, 1) density is $f(x) = 1$ for $0 \le x \le 1$, which is the Uniform(0, 1) density. The Uniform is a special Beta, and the Beta is a one-parameter generalization of the Uniform whenever the shape parameters are not both one. Some Bayesian software libraries default to a Beta(1, 1) prior; this is the noninformative-uniform-prior choice, not the Jeffreys prior.

Watch Out

The mode is not the mean

For $\alpha, \beta > 1$ the mode is $(\alpha-1)/(\alpha+\beta-2)$ and the mean is $\alpha/(\alpha+\beta)$. They coincide only when $\alpha = \beta$ (symmetric Beta). Bayesian MAP estimates report the mode; posterior-mean estimates report the mean. The two disagree under asymmetric priors and small sample sizes.

Watch Out

A density spike is not endpoint mass

When $\alpha < 1$, the Beta density diverges at $0$; when $\beta < 1$, it diverges at $1$. That does not mean $P(X=0)>0$ or $P(X=1)>0$. A proper Beta random variable is continuous on $(0,1)$ and assigns probability zero to each single point. The spike means small neighborhoods near the endpoint receive large probability relative to their width, not that the distribution has an atom at the endpoint. This distinction matters in Bayesian updates: a $\operatorname{Beta}(1/2,1/2)$ prior can put heavy density near $0$ and $1$ without declaring either value certain.

Exercises

ExerciseCore

Problem

Let $X\sim\operatorname{Beta}(3, 2)$. Compute $\mathbb{E}[X]$, $\operatorname{Var}(X)$, and the mode.

Hint

Reveal Solution

ExerciseCore

Problem

An A/B test starts with a Beta(1, 1) prior on the conversion probability $p$. After serving 200 users, 36 convert. Find the posterior distribution and the posterior mean, and compare to the MLE.

Hint

Reveal Solution

ExerciseAdvanced

Problem

Let $U_1,\dots,U_{10}$ be i.i.d. $\operatorname{Unif}(0,1)$. Identify the distribution of the median $U_{(5)}$ (with $n = 10$) and the third-largest value $U_{(8)}$.

Hint

Reveal Solution

ExerciseAdvanced

Problem

Let $X\sim\operatorname{Gamma}(\alpha,1)$ and $Y\sim\operatorname{Gamma}(\beta,1)$ be independent. Show that $X/(X+Y)\sim\operatorname{Beta}(\alpha,\beta)$, independent of $X+Y$.

Hint

Reveal Solution

References

Canonical:

• Casella and Berger, Statistical Inference (2002), Chapter 3 (Section 3.3 on Beta), Chapter 5 (Section 5.4 on order statistics).
• Lehmann and Casella, Theory of Point Estimation (1998), Chapter 4 (conjugate priors).
• Bickel and Doksum, Mathematical Statistics, Volume I (2015), Chapter 1 (conjugate families).

Bayesian framing:

• Gelman et al., Bayesian Data Analysis (2013), Chapter 2 (Section 2.4 on the Beta-Binomial model).
• Robert, The Bayesian Choice (2007), Chapter 3.
• Jaynes, Probability Theory: The Logic of Science (2003), Chapter 6 (uniform and Jeffreys priors for the Bernoulli).

Order statistics:

• David and Nagaraja, Order Statistics (2003), Chapter 2 (uniform order statistics and the Beta distribution).