{"id":2523,"date":"2026-02-02T09:53:10","date_gmt":"2026-02-02T09:53:10","guid":{"rendered":"https:\/\/demo.materiamedica.net\/demo6\/?p=2523"},"modified":"2026-02-02T09:53:10","modified_gmt":"2026-02-02T09:53:10","slug":"chapter-13-rayleigh-distribution","status":"publish","type":"post","link":"https:\/\/demo.materiamedica.net\/demo6\/chapter-13-rayleigh-distribution\/","title":{"rendered":"Chapter 13: Rayleigh Distribution"},"content":{"rendered":"

1. What is the Rayleigh distribution really?<\/h3>\n

The Rayleigh distribution<\/strong> describes the magnitude<\/strong> (absolute value \/ length) of a two-dimensional random vector<\/strong> whose components are independent normal random variables with zero mean and equal variance<\/strong>.<\/p>\n

In simpler words:<\/p>\n

\n

If you have two independent normal random variables X ~ N(0, \u03c3\u00b2) and Y ~ N(0, \u03c3\u00b2), then the distance from the origin R = \u221a(X\u00b2 + Y\u00b2) follows a Rayleigh distribution<\/strong> with scale parameter \u03c3.<\/p>\n<\/blockquote>\n

Key intuition<\/strong> (say this sentence out loud):<\/p>\n

\n

Rayleigh distribution = distribution of the distance from origin<\/strong> in 2D when you add two independent Gaussian noises with same variance.<\/p>\n<\/blockquote>\n

This situation appears very frequently<\/strong> when you measure amplitude<\/strong> or envelope<\/strong> of a signal that consists of many small random reflections \/ contributions.<\/p>\n

2. The most important properties (write these down)<\/h3>\n
\n
\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Parameter<\/th>\nSymbol<\/th>\nMeaning \/ Formula<\/th>\n<\/tr>\n<\/thead>\n
Scale parameter<\/td>\n\u03c3<\/td>\nControls spread (\u03c3 > 0)<\/td>\n<\/tr>\n
Support<\/td>\nx \u2265 0<\/td>\n<\/td>\n<\/tr>\n
Probability density<\/td>\nf(x) = (x \/ \u03c3\u00b2) \u00d7 exp(\u2212x\u00b2 \/ (2\u03c3\u00b2)) for x \u2265 0<\/td>\n<\/td>\n<\/tr>\n
Cumulative (CDF)<\/td>\nF(x) = 1 \u2212 exp(\u2212x\u00b2 \/ (2\u03c3\u00b2))<\/td>\n<\/td>\n<\/tr>\n
Mean<\/td>\nE[R] = \u03c3 \u00d7 \u221a(\u03c0\/2) \u2248 1.2533 \u03c3<\/td>\n<\/td>\n<\/tr>\n
Variance<\/td>\nVar(R) = (4\u2212\u03c0)\/2 \u00d7 \u03c3\u00b2 \u2248 0.4292 \u03c3\u00b2<\/td>\n<\/td>\n<\/tr>\n
Mode<\/td>\n\u03c3 (the peak is exactly at \u03c3)<\/td>\n<\/td>\n<\/tr>\n
Median<\/td>\n\u03c3 \u00d7 \u221a(ln 2) \u2248 1.177 \u03c3<\/td>\n<\/td>\n<\/tr>\n
Most probable value<\/td>\n\u03c3<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
<\/div>\n<\/div>\n<\/div>\n

Very useful rule of thumb<\/strong>:<\/p>\n

    \n
  • About 63%<\/strong> of values are below \u03c3 \u00d7 \u221a(ln 4) \u2248 1.177\u03c3<\/strong> (median)<\/li>\n
  • The mean is roughly 1.25\u03c3<\/strong><\/li>\n
  • Standard deviation is roughly 0.655\u03c3<\/strong><\/li>\n<\/ul>\n

    3. Generating Rayleigh random numbers in NumPy \/ SciPy<\/h3>\n
    \n
    \n
    \n
    Python<\/div>\n
    \n
    # Rayleigh with scale \u03c3 = 2 (most common way)\r\nrayleigh_data = stats.rayleigh.rvs(scale=2, size=80000)\r\n\r\nprint(\"First 10 values:\", rayleigh_data[:10].round(3))\r\nprint(\"Sample mean:\", rayleigh_data.mean().round(3))\r\nprint(\"Theoretical mean:\", 2 * np.sqrt(np.pi\/2).round(3))  # \u2248 2.5066\r\nprint(\"Sample std :\", rayleigh_data.std().round(3))\r\nprint(\"Theoretical std:\", 2 * np.sqrt((4-np.pi)\/2).round(3))  # \u2248 1.047<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    Alternative way using NumPy (very educational):<\/p>\n
    \n
    \n
    \n
    Python<\/div>\n
    \n
    sigma = 2.0\r\nX = np.random.normal(0, sigma, 100000)\r\nY = np.random.normal(0, sigma, 100000)\r\nrayleigh_manual = np.sqrt(X**2 + Y**2)\r\n\r\n# Should be almost identical to stats.rayleigh.rvs(scale=sigma)<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    4. Visualizing Rayleigh distribution (very important)<\/h3>\n
    \n
    \n
    \n
    Python<\/div>\n
    \n
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5.5))\r\n\r\nscales = [1.0, 2.0, 4.0, 8.0]\r\n\r\nx = np.linspace(0, 30, 1000)\r\n\r\nfor s in scales:\r\n    y = stats.rayleigh.pdf(x, scale=s)\r\n    ax1.plot(x, y, lw=2.3, label=f\"\u03c3 = {s}\", alpha=0.9)\r\n\r\nax1.set_title(\"Rayleigh PDF \u2013 different scale parameters\", fontsize=13)\r\nax1.set_xlabel(\"Value (r)\", fontsize=11)\r\nax1.set_ylabel(\"Density\", fontsize=11)\r\nax1.set_xlim(0, 30)\r\nax1.legend(title=\"Scale parameter \u03c3\", fontsize=11)\r\n\r\n# Cumulative distribution function (CDF)\r\nfor s in scales:\r\n    ax2.plot(x, stats.rayleigh.cdf(x, scale=s), lw=2.3, label=f\"\u03c3 = {s}\")\r\n\r\nax2.set_title(\"Rayleigh CDF\", fontsize=13)\r\nax2.set_xlabel(\"Value (r)\", fontsize=11)\r\nax2.set_ylabel(\"Cumulative probability\", fontsize=11)\r\nax2.legend(title=\"Scale parameter \u03c3\")\r\n\r\nplt.tight_layout()\r\nplt.show()<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    What you should always notice<\/strong>:<\/p>\n
      \n
    • Always starts at 0 when x=0<\/li>\n
    • Has a peak<\/strong> (mode) exactly at \u03c3<\/strong><\/li>\n
    • Never negative values<\/li>\n
    • Right-skewed, but less skewed than exponential<\/li>\n
    • Larger \u03c3 \u2192 distribution stretches to the right<\/li>\n
    • CDF always starts at 0 and approaches 1 smoothly<\/li>\n<\/ul>\n
      5. Real-world situations where Rayleigh appears naturally<\/h3>\n
      \n
      \n\n\n\n\n\n\n\n\n\n\n\n
      Domain<\/th>\nTypical use of Rayleigh distribution<\/th>\n<\/tr>\n<\/thead>\n
      Wireless communications (fading)<\/td>\nMagnitude of complex Gaussian fading channel (Rayleigh fading)<\/td>\n<\/tr>\n
      Radar \/ sonar<\/td>\nAmplitude of noise \/ target echo in non-coherent detection<\/td>\n<\/tr>\n
      Acoustics \/ audio processing<\/td>\nEnvelope of narrowband Gaussian noise<\/td>\n<\/tr>\n
      MRI \/ medical imaging<\/td>\nNoise magnitude in magnitude images<\/td>\n<\/tr>\n
      Wind speed modeling<\/td>\nSpeed of wind (often Rayleigh or Weibull)<\/td>\n<\/tr>\n
      Line-of-sight + multipath signals<\/td>\nReceived signal strength when no dominant path<\/td>\n<\/tr>\n
      Optical scattering<\/td>\nIntensity of scattered light<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
      <\/div>\n<\/div>\n<\/div>\n
      Most famous appearance<\/strong> \u2014 Rayleigh fading<\/strong> in wireless communications<\/p>\n
      In mobile communication, the received signal amplitude often follows Rayleigh when there is no line-of-sight<\/strong> path and many small reflections arrive from different directions.<\/p>\n
      6. Realistic code patterns you will actually write<\/h3>\n
      Pattern 1 \u2013 Simulate Rayleigh fading channel coefficients<\/strong><\/p>\n
      \n
      \n
      \n
      Python<\/div>\n
      \n
      # Wireless channel \u2013 Rayleigh fading with average power = 1\r\nsigma = 1 \/ np.sqrt(2)          # because Var = 2\u03c3\u00b2 = 1 \u2192 \u03c3 = 1\/\u221a2\r\n\r\nchannel_real = np.random.normal(0, sigma, 100000)\r\nchannel_imag = np.random.normal(0, sigma, 100000)\r\nchannel_amplitude = np.sqrt(channel_real**2 + channel_imag**2)\r\n\r\nprint(\"Average amplitude:\", channel_amplitude.mean().round(4))  # \u2248 1.253\r\nprint(\"Theoretical mean:\", sigma * np.sqrt(np.pi).round(4))     # \u2248 1.253<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      Pattern 2 \u2013 Simulate signal + Rayleigh fading + noise<\/strong><\/p>\n
      \n
      \n
      \n
      Python<\/div>\n
      \n
      # Transmitted signal amplitude = 2.0\r\nsignal = 2.0\r\n\r\n# Rayleigh fading\r\nh = stats.rayleigh.rvs(scale=1, size=50000)\r\n\r\n# Additive Gaussian noise (SNR simulation)\r\nnoise_power = 0.1\r\nnoise = np.random.normal(0, np.sqrt(noise_power), 50000)\r\n\r\nreceived = h * signal + noise\r\n\r\nreceived_amplitude = np.abs(received)\r\n\r\nsns.histplot(received_amplitude, bins=80, stat=\"density\", kde=True,\r\n             color=\"teal\", alpha=0.7)\r\nplt.title(\"Received signal amplitude (Rayleigh fading + noise)\")\r\nplt.xlabel(\"Amplitude\")\r\nplt.show()<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      Pattern 3 \u2013 Probability that amplitude exceeds a threshold<\/strong><\/p>\n
      \n
      \n
      \n
      Python<\/div>\n
      \n
      sigma = 1.5\r\nthreshold = 4.0\r\n\r\n# P(R > threshold) = exp(-threshold\u00b2 \/ (2\u03c3\u00b2))\r\ntheoretical_prob = np.exp(-threshold**2 \/ (2 * sigma**2))\r\n\r\n# Simulate\r\nsamples = stats.rayleigh.rvs(scale=sigma, size=100000)\r\nempirical_prob = np.mean(samples > threshold)\r\n\r\nprint(f\"Theoretical P(R > {threshold}) = {theoretical_prob:.5f}\")\r\nprint(f\"Empirical P(R > {threshold})    = {empirical_prob:.5f}\")<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      Summary \u2013 Rayleigh Distribution Quick Reference<\/h3>\n
      \n
      \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
      Property<\/th>\nValue \/ Formula<\/th>\n<\/tr>\n<\/thead>\n
      Shape<\/td>\nRight-skewed, peak at \u03c3, no left tail<\/td>\n<\/tr>\n
      Defined by<\/td>\nscale parameter \u03c3 (\u03c3 > 0)<\/td>\n<\/tr>\n
      Mean<\/td>\n\u03c3 \u00d7 \u221a(\u03c0\/2) \u2248 1.2533 \u03c3<\/td>\n<\/tr>\n
      Variance<\/td>\n(4\u2212\u03c0)\/2 \u00d7 \u03c3\u00b2 \u2248 0.4292 \u03c3\u00b2<\/td>\n<\/tr>\n
      Mode<\/td>\n\u03c3 (peak exactly at \u03c3)<\/td>\n<\/tr>\n
      Median<\/td>\n\u03c3 \u00d7 \u221a(ln 2) \u2248 1.177 \u03c3<\/td>\n<\/tr>\n
      Support<\/td>\nx \u2265 0<\/td>\n<\/tr>\n
      PDF<\/td>\n(x \/ \u03c3\u00b2) exp(\u2212x\u00b2 \/ (2\u03c3\u00b2)) for x \u2265 0<\/td>\n<\/tr>\n
      CDF<\/td>\n1 \u2212 exp(\u2212x\u00b2 \/ (2\u03c3\u00b2))<\/td>\n<\/tr>\n
      NumPy \/ SciPy<\/td>\nscipy.stats.rayleigh.rvs(scale=\u03c3, size=…)<\/td>\n<\/tr>\n
      Most common use cases<\/td>\nSignal envelope, fading channels, radar cross-section, magnitude of 2D Gaussian<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
      <\/div>\n<\/div>\n<\/div>\n
      Final teacher messages<\/h3>\n
        \n
      1. Whenever you are looking at the magnitude \/ amplitude \/ envelope of a signal<\/strong> that comes from two independent Gaussian components \u2192 think Rayleigh.<\/li>\n
      2. Rayleigh = special case of Rice distribution<\/strong> when there is no line-of-sight component (non-centrality = 0).<\/li>\n
      3. Rayleigh fading<\/strong> is one of the most famous applications \u2014 you will see it constantly if you work in wireless communications or signal processing.<\/li>\n<\/ol>\n
        Would you like to continue with any of these next?<\/p>\n
          \n
        • Rayleigh vs Rician distribution (when there is a dominant path)<\/li>\n
        • How to estimate \u03c3 from real data<\/li>\n
        • Rayleigh fading simulation for communication systems<\/li>\n
        • Realistic mini-project: simulate signal strength + fading + noise<\/li>\n
        • Comparing Rayleigh with exponential, chi-square, gamma<\/li>\n<\/ul>\n
          Just tell me what you want to explore next! \ud83d\ude0a<\/p>\n","protected":false},"excerpt":{"rendered":"
          1. What is the Rayleigh distribution really? The Rayleigh distribution describes the magnitude (absolute value \/ length) of a two-dimensional random vector whose components are independent normal random variables with zero mean and equal...<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[75],"tags":[],"class_list":["post-2523","post","type-post","status-publish","format-standard","hentry","category-numpy"],"_links":{"self":[{"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/posts\/2523","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/comments?post=2523"}],"version-history":[{"count":1,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/posts\/2523\/revisions"}],"predecessor-version":[{"id":2524,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/posts\/2523\/revisions\/2524"}],"wp:attachment":[{"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/media?parent=2523"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/categories?post=2523"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/tags?post=2523"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}