Moments and Stats¶

The moments of a probability distribution encode its shape: the mean locates the center, the variance measures spread, skewness quantifies asymmetry, and kurtosis captures tail weight. Rather than estimating these from samples, scipy.stats distribution objects compute them analytically from the distribution's parameters, giving exact values with no sampling error. This page defines each moment and demonstrates the corresponding scipy.stats methods.

Mean and Variance¶

The mean (first moment about the origin) of a continuous random variable \(X\) with PDF \(f\) is:

The variance (second central moment) measures the expected squared deviation from the mean:

The standard deviation is \(\sigma = \sqrt{\sigma^2}\).

Computing Mean, Variance, and Standard Deviation¶

```python import scipy.stats as stats

Normal distribution with μ=5, σ=2¶

dist = stats.norm(loc=5, scale=2)

print(f"Mean: {dist.mean():.4f}") # 5.0 print(f"Variance: {dist.var():.4f}") # 4.0 print(f"Std dev: {dist.std():.4f}") # 2.0 ```

These methods return exact analytical values. For the normal distribution, .mean() returns loc, .var() returns scale**2, and .std() returns scale.

Example: Gamma Distribution¶

The gamma distribution with shape \(a\) and scale \(\theta\) has mean \(a\theta\) and variance \(a\theta^2\):

python dist = stats.gamma(a=3, scale=2) print(f"Mean: {dist.mean():.4f}") # 6.0 (3 * 2) print(f"Variance: {dist.var():.4f}") # 12.0 (3 * 4)

Skewness and Kurtosis¶

The skewness measures asymmetry of the distribution about its mean:

A positive value indicates a right tail longer than the left; a negative value indicates the opposite. A symmetric distribution has \(\gamma_1 = 0\).

The excess kurtosis measures the heaviness of the tails relative to the normal distribution:

The subtraction of 3 makes the normal distribution the reference point with \(\gamma_2 = 0\). Positive excess kurtosis (leptokurtic) indicates heavier tails; negative (platykurtic) indicates lighter tails.

The .stats() Method¶

The .stats() method returns multiple moments in a single call. The moments keyword controls which are computed, using a string of characters:

```python dist = stats.norm(loc=0, scale=1) mean, var, skew, kurt = dist.stats(moments='mvsk') print(f"Mean={mean:.2f}, Var={var:.2f}, Skew={skew:.2f}, Kurt={kurt:.2f}")

Mean=0.00, Var=1.00, Skew=0.00, Kurt=0.00¶

```

Example: Exponential Distribution¶

The exponential distribution with rate \(\lambda\) (scale \(= 1/\lambda\)) is right-skewed:

```python dist = stats.expon(scale=1.0) mean, var, skew, kurt = dist.stats(moments='mvsk') print(f"Mean={mean:.2f}, Var={var:.2f}, Skew={skew:.2f}, Kurt={kurt:.2f}")

Mean=1.00, Var=1.00, Skew=2.00, Kurt=6.00¶

```

The skewness of 2 confirms the strong right skew, and the excess kurtosis of 6 indicates substantially heavier tails than the normal.

The .moment() Method¶

The .moment(n) method computes the \(n\)-th non-central moment \(\mathbb{E}[X^n]\):

```python dist = stats.norm(loc=0, scale=1)

First non-central moment = mean¶

print(f"E[X] = {dist.moment(1):.4f}") # 0.0

Second non-central moment = E[X^2] = Var + mean^2¶

print(f"E[X^2] = {dist.moment(2):.4f}") # 1.0

Fourth non-central moment¶

print(f"E[X^4] = {dist.moment(4):.4f}") # 3.0 ```

The .entropy() Method¶

The differential entropy of a continuous distribution measures its uncertainty:

```python

Normal distribution: entropy = 0.5 * ln(2πeσ²)¶

dist = stats.norm(loc=0, scale=1) print(f"Entropy: {dist.entropy():.4f}") # 1.4189 ```

Higher entropy indicates greater spread or uncertainty in the distribution.

Method Summary¶

Summary¶

The moments of a distribution — mean, variance, skewness, and kurtosis — summarize its center, spread, asymmetry, and tail behavior. In scipy.stats, the .mean(), .var(), .std(), and .stats() methods compute these analytically from the distribution parameters, while .moment(n) provides arbitrary non-central moments. These exact values serve as benchmarks when comparing against sample estimates computed from data.

Exercises¶

Exercise 1. For an exponential distribution with \(\lambda = 0.5\) (i.e., scale=2), compute the mean, variance, skewness, and kurtosis using .stats(moments='mvsk'). Verify the mean equals \(1/\lambda\).

from scipy import stats

rv = stats.expon(scale=2)
m, v, s, k = rv.stats(moments='mvsk')
print(f"Mean: {float(m):.4f} (expected {1/0.5:.4f})")
print(f"Variance: {float(v):.4f}")
print(f"Skewness: {float(s):.4f}")
print(f"Kurtosis: {float(k):.4f}")

Exercise 2. Compute the differential entropy of a normal distribution for \(\sigma = 1, 2, 5\) using the .entropy() method. Verify that the entropy of \(N(0, \sigma^2)\) equals \(\frac{1}{2}\ln(2\pi e \sigma^2)\).

import numpy as np
from scipy import stats

for sigma in [1, 2, 5]:
    rv = stats.norm(scale=sigma)
    entropy = rv.entropy()
    expected = 0.5 * np.log(2 * np.pi * np.e * sigma**2)
    print(f"sigma={sigma}: entropy={entropy:.4f}, expected={expected:.4f}, "
          f"match={np.isclose(entropy, expected)}")

Exercise 3. Use .moment(n) to compute the 1st through 4th non-central moments of a standard normal distribution. Verify that odd moments are zero and the 2nd moment equals 1.

import numpy as np
from scipy import stats

rv = stats.norm()
for n in range(1, 5):
    moment_val = rv.moment(n)
    print(f"E[X^{n}] = {moment_val:.4f}")