41  Power Distribution

The Power distribution is a one-parameter family on \([0,1]\) that generalizes the Uniform distribution. A single shape parameter tilts the density toward 0 (for \(\alpha < 1\)) or toward 1 (for \(\alpha > 1\)), making it useful for bounded quantities with a directional preference.

Formally, the random variate \(X\) defined for the range \(X \in [0, 1]\), is said to have a Power Distribution (i.e. \(X \sim \text{Power}(\alpha)\)) with shape parameter \(\alpha > 0\). The Power distribution is equivalent to Beta\((\alpha, 1)\).

41.1 Probability Density Function

\[ f(x) = \alpha\, x^{\alpha-1}, \quad 0 \leq x \leq 1 \]

The figure below shows examples of the Power Probability Density Function for different shape values.

Code 
par(mfrow = c(2, 2))
x <- seq(0, 1, length = 500)

plot(x, dbeta(x, shape1 = 0.5, shape2 = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(alpha == 0.5))

plot(x, dbeta(x, shape1 = 1, shape2 = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(alpha == 1))

plot(x, dbeta(x, shape1 = 2, shape2 = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(alpha == 2))

plot(x, dbeta(x, shape1 = 5, shape2 = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(alpha == 5))

par(mfrow = c(1, 1))
Figure 41.1: Power Probability Density Function for various shape values

41.2 Purpose

The Power distribution models bounded quantities in \([0,1]\) when the density is monotone — either increasing or decreasing — rather than bell-shaped. Its single parameter determines whether small values (near 0) or large values (near 1) are most likely. Common applications include:

  • Proportion of correct answers when most respondents score high or low
  • Reliability: proportion of items surviving a stress test when failure is rare or common
  • Revenue share distributions: when one party captures most of the total
  • Prior distribution for probabilities biased toward extreme values
  • Order statistics from the Uniform distribution on \([0,1]\)

Relation to the discrete setting. Power\((\alpha)\) = Beta\((\alpha, 1)\) is a continuous model for proportions; its discrete analog is the Bernoulli distribution (for a single bounded outcome) or a Discrete Uniform (when \(\alpha = 1\)). For integer-valued bounded counts, the Binomial plays a similar role.

41.3 Distribution Function

\[ F(x) = x^\alpha, \quad 0 \leq x \leq 1 \]

The figure below shows the Power Distribution Function for \(\alpha = 3\).

Code 
x <- seq(0, 1, length = 500)
plot(x, pbeta(x, shape1 = 3, shape2 = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "F(x)", main = "Power Distribution Function",
     sub = expression(alpha == 3))
Figure 41.2: Power Distribution Function (alpha = 3)

41.4 Moment Generating Function

The MGF has no simple closed form. Raw moments are computed directly:

\[ \mu_n' = \frac{\alpha}{\alpha + n} \]

41.5 1st Uncentered Moment

\[ \mu_1' = \frac{\alpha}{\alpha + 1} \]

41.6 2nd Uncentered Moment

\[ \mu_2' = \frac{\alpha}{\alpha + 2} \]

41.7 3rd Uncentered Moment

\[ \mu_3' = \frac{\alpha}{\alpha + 3} \]

41.8 4th Uncentered Moment

\[ \mu_4' = \frac{\alpha}{\alpha + 4} \]

41.9 2nd Centered Moment

\[ \mu_2 = \frac{\alpha}{(\alpha+1)^2(\alpha+2)} \]

41.10 3rd Centered Moment

\[ \mu_3 = \frac{2\alpha(1-\alpha)}{(\alpha+1)^3(\alpha+2)(\alpha+3)} \]

41.11 4th Centered Moment

Obtained by expanding from raw moments.

41.12 Expected Value

\[ \text{E}(X) = \frac{\alpha}{\alpha + 1} \]

41.13 Variance

\[ \text{V}(X) = \frac{\alpha}{(\alpha+1)^2(\alpha+2)} \]

41.14 Median

\[ \text{Med}(X) = 2^{-1/\alpha} \]

41.15 Mode

\[ \text{Mo}(X) = \begin{cases} 1 & \alpha > 1 \\ 0 & \alpha < 1 \\ \text{any value in } [0,1] & \alpha = 1 \end{cases} \]

41.16 Coefficient of Skewness

\[ g_1 = \frac{2(1-\alpha)\sqrt{\alpha+2}}{(\alpha+3)\sqrt{\alpha}} \]

For \(\alpha = 1\): \(g_1 = 0\) (Uniform is symmetric). For \(\alpha > 1\): \(g_1 < 0\) (left-skewed toward 1).

41.17 Coefficient of Kurtosis

\[ g_2 = 3 + \frac{6(\alpha^3 - \alpha^2 - 2\alpha - 2)}{\alpha(\alpha+3)(\alpha+4)} \]

At \(\alpha = 1\) (Uniform): \(g_2 = 3 + 6(1-1-2-2)/(1\cdot 4\cdot 5) = 3 - 24/20 = 3 - 1.2 = 1.8\). ✓

41.18 Parameter Estimation

The MLE has a rare closed form:

\[ \hat\alpha = -\frac{n}{\displaystyle\sum_{i=1}^n \ln x_i} \]

set.seed(42)
alpha_true <- 3
x_obs <- rbeta(100, shape1 = alpha_true, shape2 = 1)

# MLE
alpha_hat <- -length(x_obs) / sum(log(x_obs))
cat("MLE alpha:", round(alpha_hat, 4), "\n")
cat("True alpha:", alpha_true, "\n")

# Mean comparison
cat("MLE mean:", alpha_hat / (alpha_hat + 1), "\n")
cat("Sample mean:", mean(x_obs), "\n")
MLE alpha: 2.9641 
True alpha: 3 
MLE mean: 0.7477371 
Sample mean: 0.7450839

41.19 R Module

41.19.1 RFC

The Power Distribution module is available in RFC under the menu “Distributions / Power Distribution”.

41.19.3 R Code

The following code demonstrates Power distribution probability calculations using dbeta/pbeta with shape2 = 1:

alpha <- 3

# Using Beta(alpha, 1) equivalence
dbeta(0.75, shape1 = alpha, shape2 = 1)

# P(X <= 0.75)
pbeta(0.75, shape1 = alpha, shape2 = 1)

# Median
cat("Median:", 2^(-1/alpha), "\n")

# Mean
cat("Mean:", alpha / (alpha + 1), "\n")
[1] 1.6875
[1] 0.421875
Median: 0.7937005 
Mean: 0.75

41.20 Example

The proportion of correct answers on a test is modeled as \(X \sim \text{Power}(3)\). This implies the density is \(f(x) = 3x^2\), concentrated toward high scores. The mean proportion is \(3/4 = 0.75\).

alpha <- 3

# P(proportion <= 0.75)
cat("P(score proportion <= 0.75):", pbeta(0.75, alpha, 1), "\n")

# Mean and median
cat("Mean:", alpha / (alpha + 1), "\n")
cat("Median:", round(2^(-1/alpha), 4), "\n")

# Skewness (negative = left-skewed toward 1)
g1 <- 2 * (1 - alpha) * sqrt(alpha + 2) / ((alpha + 3) * sqrt(alpha))
cat("Skewness g1:", round(g1, 4), "\n")
P(score proportion <= 0.75): 0.421875 
Mean: 0.75 
Median: 0.7937 
Skewness g1: -0.8607
Interactive Shiny app (click to load).
Open in new tab

41.21 Random Number Generator

The Power distribution has the simplest possible random number generator via the inverse-CDF. Since \(F(x) = x^\alpha\):

\[ X = U^{1/\alpha} \sim \text{Power}(\alpha) \quad \text{when } U \sim \text{U}(0,1) \]

set.seed(123)
n <- 1000; alpha <- 3

# Inverse-transform method
u <- runif(n)
x_inv <- u^(1/alpha)

cat("Simulated mean:", round(mean(x_inv), 4), "\n")
cat("Theoretical mean:", alpha / (alpha + 1), "\n")
cat("Simulated var:", round(var(x_inv), 4), "\n")
cat("Theoretical var:", alpha / ((alpha+1)^2 * (alpha+2)), "\n")
Simulated mean: 0.7487 
Theoretical mean: 0.75 
Simulated var: 0.0374 
Theoretical var: 0.0375
Interactive Shiny app (click to load).
Open in new tab

41.22 Property 1: Special Case of Beta

The Power distribution is the Beta distribution with shape parameters \((\alpha, 1)\):

\[ \text{Power}(\alpha) = \text{Beta}(\alpha, 1) \]

All Beta distribution properties apply, and R’s dbeta, pbeta, qbeta, and rbeta with shape2 = 1 compute the Power distribution exactly. See Chapter 30.

41.23 Property 2: Uniform Special Case

When \(\alpha = 1\), the Power distribution reduces to the Uniform\((0, 1)\) distribution:

\[ \text{Power}(1) = \text{U}(0, 1) \]

See Chapter 19.

41.24 Property 3: Closed-Form MLE

The MLE \(\hat\alpha = -n/\sum \ln x_i\) is one of the few analytically tractable maximum likelihood estimators in the Beta family, but it is biased for \(n > 1\). For \(n > 1\), the UMVUE is \(\hat\alpha_U = (n-1)/(-\sum \ln x_i)\).