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sdr.subtract_rvs(X: rv_continuous | rv_histogram, Y: rv_continuous | rv_histogram, p: float =
1e-16
) rv_histogram Numerically calculates the distribution of the difference of two independent random variables \(X\) and \(Y\).
- Parameters:¶
- X: rv_continuous | rv_histogram¶
The distribution of the random variable \(X\).
- Y: rv_continuous | rv_histogram¶
The distribution of the 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 difference \(Z = X - Y\).
Notes
Given two independent random variables \(X\) and \(Y\) with PDFs \(f_X(x)\) and \(f_Y(y)\), we compute the PDF of \(Z = X - Y\) as follows.
The PDF of \(Z\), denoted \(f_Z(z)\), can be obtained using the convolution formula for independent random variables. For the difference \(Z = X - Y\), the PDF of \(Y\) is flipped.
\[f_Z(z) = \int_{-\infty}^\infty f_X(x) f_Y(x - z) \, dx\]Examples
Compute the distribution of the difference of normal and Rayleigh random variables.
In [1]: X = scipy.stats.norm(loc=5, scale=0.5) In [2]: Y = scipy.stats.rayleigh(loc=1, scale=2) 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.subtract_rvs(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("Difference of normal and Rayleigh random variables"); ...:
Compute the distribution of the difference of Rayleigh and Rician random variables.
In [5]: X = scipy.stats.rayleigh(scale=1) In [6]: Y = scipy.stats.rice(3) In [7]: x = np.linspace(-10, 10, 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.subtract_rvs(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("Difference of Rayleigh and Rician random variables"); ...:
Compute the distribution of the difference of Rician and Chi-squared random variables.
In [9]: X = scipy.stats.rice(3) In [10]: Y = scipy.stats.chi2(3) In [11]: x = np.linspace(-10, 10, 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.subtract_rvs(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.xlim(-10, 10); \ ....: plt.xlabel("Random variable"); \ ....: plt.ylabel("Probability density"); \ ....: plt.title("Difference of Rician and Chi-squared random variables"); ....: