sdr.sum_distributions(X: rv_continuous | rv_histogram, Y: rv_continuous | rv_histogram, p: float = 1e-16) → rv_histogram

Numerically calculates the distribution of the sum of two independent random variables \(X\) and \(Y\).

Parameters:
X: rv_continuous | rv_histogram

The distribution of the first random variable \(X\).

Y: rv_continuous | rv_histogram

The distribution of the second random variable \(Y\).

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 \(Z = X + Y\).

Notes

The PDF of the sum of two independent random variables is the convolution of the PDF of the two distributions.

\[f_{X+Y}(t) = (f_X * f_Y)(t)\]

Examples

Compute the distribution of the sum of two normal distributions.

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

In [2]: Y = scipy.stats.norm(loc=2, scale=1.5)

In [3]: x = np.linspace(-5, 10, 1_001)

In [4]: plt.figure(); \
   ...: plt.plot(x, X.pdf(x), label="X"); \
   ...: plt.plot(x, Y.pdf(x), label="Y"); \
   ...: plt.plot(x, sdr.sum_distributions(X, Y).pdf(x), label="X + Y"); \
   ...: plt.hist(X.rvs(100_000) + Y.rvs(100_000), bins=101, density=True, histtype="step", label="X + Y empirical"); \
   ...: plt.legend(); \
   ...: plt.xlabel("Random variable"); \
   ...: plt.ylabel("Probability density"); \
   ...: plt.title("Sum of two Normal distributions");
   ...:
../../_images/sdr_sum_distributions_1.png

Compute the distribution of the sum of two Rayleigh distributions.

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

In [6]: Y = scipy.stats.rayleigh(loc=1, scale=2)

In [7]: x = np.linspace(0, 12, 1_001)

In [8]: plt.figure(); \
   ...: plt.plot(x, X.pdf(x), label="X"); \
   ...: plt.plot(x, Y.pdf(x), label="Y"); \
   ...: plt.plot(x, sdr.sum_distributions(X, Y).pdf(x), label="X + Y"); \
   ...: plt.hist(X.rvs(100_000) + Y.rvs(100_000), bins=101, density=True, histtype="step", label="X + Y empirical"); \
   ...: plt.legend(); \
   ...: plt.xlabel("Random variable"); \
   ...: plt.ylabel("Probability density"); \
   ...: plt.title("Sum of two Rayleigh distributions");
   ...:
../../_images/sdr_sum_distributions_2.png

Compute the distribution of the sum of two Rician distributions.

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

In [10]: Y = scipy.stats.rice(3)

In [11]: x = np.linspace(0, 12, 1_001)

In [12]: plt.figure(); \
   ....: plt.plot(x, X.pdf(x), label="X"); \
   ....: plt.plot(x, Y.pdf(x), label="Y"); \
   ....: plt.plot(x, sdr.sum_distributions(X, Y).pdf(x), label="X + Y"); \
   ....: plt.hist(X.rvs(100_000) + Y.rvs(100_000), bins=101, density=True, histtype="step", label="X + Y empirical"); \
   ....: plt.legend(); \
   ....: plt.xlabel("Random variable"); \
   ....: plt.ylabel("Probability density"); \
   ....: plt.title("Sum of two Rician distributions");
   ....:
../../_images/sdr_sum_distributions_3.png