sdr.add_iid_rvs

sdr.add_iid_rvs(X: rv_continuous | rv_histogram, n_vars: int, p: float = 1e-16) → rv_histogram

Numerically calculates the distribution of the sum of $n$ i.i.d. random variables $X_{i}$ .

Parameters:¶

X: rv_continuous | rv_histogram¶: The distribution of the i.i.d. random variables $X_{i}$ .
n_vars: int¶: The number $n$ of random variables.
p: float = 1e-16¶: The probability of exceeding the x axis, on either side, for each distribution. This is used to determine the bounds on the x axis for the numerical convolution. Smaller values of $p$ will result in more accurate analysis, but will require more computation.

Returns:¶

The distribution of the sum $Z = X_{1} + X_{2} + \dots + X_{n}$ .

Notes¶

Given a random variable $X$ with PDF $f_{X} (x)$ , we compute the PDF of $Z = X_{1} + X_{2} + \dots + X_{n}$ , where $X_{1}, X_{2}, \dots, X_{n}$ are independent and identically distributed (i.i.d.), as follows.

The PDF of $Z$ , denoted $f_{Z} (z)$ , can be obtained by using the convolution formula for independent random variables. Specifically, the PDF of the sum of $n$ i.i.d. random variables is given by the $n$ -fold convolution of the PDF of $X$ with itself.

For $n = 2$ , $Z = X_{1} + X_{2}$ . The PDF of $Z$ is

f_{Z} (z) = \int_{- \infty}^{\infty} f_{X} (x) f_{X} (z - x) d x

For $n > 2$ , the PDF of $Z = X_{1} + X_{2} + \dots + X_{n}$ is computed recursively

f_{Z} (z) = \int_{- \infty}^{\infty} f_{X} (x) f_{Z_{n - 1}} (z - x) d x

where $f_{Z_{n - 1}} (z)$ is the PDF of the sum of $n - 1$ random variables.

For large $n$ , the Central Limit Theorem may be used as an approximation. If $X_{i}$ have mean $μ$ and variance $σ^{2}$ , then $Z$ approximately follows $Z \sim N (n μ, n σ^{2})$ for sufficiently large $n$ .

Examples¶

Compute the distribution of the sum of two normal random variables.

In [1]: X = scipy.stats.norm(loc=-1, scale=0.5)

In [2]: n_vars = 2

In [3]: x = np.linspace(-6, 2, 1_001)

In [4]: plt.figure(); \
   ...: plt.plot(x, X.pdf(x), label="$X$"); \
   ...: plt.plot(x, sdr.add_iid_rvs(X, n_vars).pdf(x), label="$X_1 + X_2$"); \
   ...: plt.hist(X.rvs((100_000, n_vars)).sum(axis=1), bins=101, density=True, histtype="step", label="$X_1 + X_2$ empirical"); \
   ...: plt.legend(); \
   ...: plt.xlabel("Random variable"); \
   ...: plt.ylabel("Probability density"); \
   ...: plt.title("Sum of two normal random variables");
   ...: 

Compute the distribution of the sum of three Rayleigh random variables.

In [5]: X = scipy.stats.rayleigh(scale=1)

In [6]: n_vars = 3

In [7]: x = np.linspace(0, 10, 1_001)

In [8]: plt.figure(); \
   ...: plt.plot(x, X.pdf(x), label="$X$"); \
   ...: plt.plot(x, sdr.add_iid_rvs(X, n_vars).pdf(x), label="$X_1 + X_2 + X_3$"); \
   ...: plt.hist(X.rvs((100_000, n_vars)).sum(axis=1), bins=101, density=True, histtype="step", label="$X_1 + X_2 + X_3$ empirical"); \
   ...: plt.legend(); \
   ...: plt.xlabel("Random variable"); \
   ...: plt.ylabel("Probability density"); \
   ...: plt.title("Sum of three Rayleigh random variables");
   ...: 

Compute the distribution of the sum of four Rician random variables.

In [9]: X = scipy.stats.rice(2)

In [10]: n_vars = 4

In [11]: x = np.linspace(0, 18, 1_001)

In [12]: plt.figure(); \
   ....: plt.plot(x, X.pdf(x), label="$X$"); \
   ....: plt.plot(x, sdr.add_iid_rvs(X, n_vars).pdf(x), label="$X_1 + X_2 + X_3 + X_4$"); \
   ....: plt.hist(X.rvs((100_000, n_vars)).sum(axis=1), bins=101, density=True, histtype="step", label="$X_1 + X_2 + X_3 + X_4$ empirical"); \
   ....: plt.legend(); \
   ....: plt.xlabel("Random variable"); \
   ....: plt.ylabel("Probability density"); \
   ....: plt.title("Sum of four Rician random variables");
   ....: 