{"id":2512,"date":"2026-02-02T09:39:59","date_gmt":"2026-02-02T09:39:59","guid":{"rendered":"https:\/\/demo.materiamedica.net\/demo6\/?p=2512"},"modified":"2026-02-02T09:39:59","modified_gmt":"2026-02-02T09:39:59","slug":"chapter-8-uniform-distribution","status":"publish","type":"post","link":"https:\/\/demo.materiamedica.net\/demo6\/chapter-8-uniform-distribution\/","title":{"rendered":"Chapter 8: Uniform Distribution"},"content":{"rendered":"

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

The uniform distribution<\/strong> is the simplest and most intuitive continuous probability distribution.<\/p>\n

It means:<\/p>\n

\n

Every value inside a certain interval is equally likely<\/strong> to occur.<\/p>\n<\/blockquote>\n

There are no favorite values<\/strong> \u2014 no peak, no bell shape, no skew. It\u2019s completely flat between the minimum and maximum.<\/p>\n

There are two main flavors we care about:<\/p>\n

    \n
  • Continuous uniform<\/strong> \u2192 any real number between a and b is equally likely<\/li>\n
  • Discrete uniform<\/strong> \u2192 equally likely integers (like rolling a fair die)<\/li>\n<\/ul>\n

    Key parameters<\/strong> (only two):<\/p>\n

      \n
    • a<\/strong> = minimum value (lower bound)<\/li>\n
    • b<\/strong> = maximum value (upper bound)<\/li>\n<\/ul>\n

      The probability density is constant: f(x) = 1 \/ (b \u2212 a)<\/strong> for a \u2264 x \u2264 b f(x) = 0<\/strong> everywhere else<\/p>\n

      2. Mental picture \u2014 how it looks<\/h3>\n

      Imagine you drop a dart completely randomly on a number line between 10 and 30. Every single point between 10 and 30 has exactly the same chance of being hit.<\/p>\n

      The histogram looks like a perfect rectangle<\/strong>:<\/p>\n

      \n
      \n
      \n
      text<\/div>\n
      \n
      \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\r\n   \u2502                             \u2502\r\n   \u2502                             \u2502\r\n   \u2502                             \u2502   \u2190 flat = uniform\r\n   \u2502                             \u2502\r\n   \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518\r\n      10                        30<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      3. Generating uniform random numbers in NumPy<\/h3>\n
      NumPy gives you two very commonly used functions:<\/p>\n
      A. np.random.rand() \u2014 standard uniform [0, 1)<\/h4>\n
      \n
      \n
      \n
      Python<\/div>\n
      \n
      # Single number\r\nprint(np.random.rand())\r\n\r\n# 1D array\r\nprint(np.random.rand(8))\r\n\r\n# 2D array \u2014 very common shape\r\nprint(np.random.rand(5, 6))\r\n\r\n# Large array for visualization\r\nu = np.random.rand(100000)<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      B. np.random.uniform(low, high, size=…) \u2014 custom interval<\/h4>\n
      \n
      \n
      \n
      Python<\/div>\n
      \n
      # Between 0 and 100 (like random percentages)\r\nprint(np.random.uniform(0, 100, 10).round(1))\r\n\r\n# Between -5 and 5 (symmetric around zero)\r\nsymmetric = np.random.uniform(-5, 5, 50000)\r\n\r\n# Temperatures between 18\u00b0C and 32\u00b0C\r\ndaily_temps = np.random.uniform(18, 32, 365)<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      4. Visualizing the uniform distribution (very important)<\/h3>\n
      \n
      \n
      \n
      Python<\/div>\n
      \n
      fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))\r\n\r\n# Standard uniform [0,1)\r\nsns.histplot(np.random.rand(80000), bins=60, stat=\"density\",\r\n             color=\"skyblue\", alpha=0.8, ax=ax1)\r\nax1.set_title(\"Uniform distribution [0, 1)\", fontsize=13)\r\nax1.set_xlabel(\"Value\")\r\nax1.set_ylabel(\"Density\")\r\nax1.set_xlim(0, 1)\r\n\r\n# Custom uniform [15, 45]\r\ncustom = np.random.uniform(15, 45, 80000)\r\nsns.histplot(custom, bins=60, stat=\"density\",\r\n             color=\"teal\", alpha=0.8, ax=ax2)\r\nax2.set_title(\"Uniform distribution [15, 45]\", fontsize=13)\r\nax2.set_xlabel(\"Value\")\r\nax2.set_ylabel(\"Density\")\r\n\r\nplt.tight_layout()\r\nplt.show()<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      What you should always see<\/strong>:<\/p>\n
        \n
      • Flat top (almost perfect rectangle when sample size is large)<\/li>\n
      • Sharp drop to zero outside [a, b]<\/li>\n
      • Density height = 1 \/ (b \u2212 a)<\/strong><\/li>\n<\/ul>\n
        5. Very common real-world use cases (you will meet these often)<\/h3>\n
        Use case 1 \u2013 Generating random test values \/ mock data<\/h4>\n
        \n
        \n
        \n
        Python<\/div>\n
        \n
        # Random prices between 9.99 and 99.99\r\nprices = np.random.uniform(9.99, 99.99, size=500)\r\n\r\n# Random percentages \/ probabilities\r\nprobabilities = np.random.rand(1000)   # [0,1)<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        Use case 2 \u2013 Random initialization \/ starting points<\/h4>\n
        \n
        \n
        \n
        Python<\/div>\n
        \n
        # Random starting weights for simple models\r\nweights = np.random.uniform(-0.1, 0.1, size=(100, 30))\r\n\r\n# Random split points for time series cross-validation\r\nsplit_points = np.random.uniform(0.1, 0.9, size=50)<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        Use case 3 \u2013 Simulating uniform sensor readings \/ background noise<\/h4>\n
        \n
        \n
        \n
        Python<\/div>\n
        \n
        # Background noise in some measurement device (uniform between -2 and +2)\r\nnoise = np.random.uniform(-2, 2, size=10000)\r\n\r\nsignal = 50 + noise<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        Use case 4 \u2013 Random sampling of categories \/ discrete uniform<\/h4>\n
        \n
        \n
        \n
        Python<\/div>\n
        \n
        options = ['A', 'B', 'C', 'D', 'E']\r\nrandom_choices = np.random.choice(options, size=10000)\r\n\r\nsns.countplot(x=random_choices, palette=\"Set2\")\r\nplt.title(\"Discrete uniform \u2013 equal probability\")\r\nplt.show()<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        6. Important properties & formulas (write these down)<\/h3>\n
        \n
        \n\n\n\n\n\n\n\n\n\n\n
        Property<\/th>\nValue \/ Formula<\/th>\n<\/tr>\n<\/thead>\n
        Range<\/td>\n[a, b] or [0, 1) for rand()<\/td>\n<\/tr>\n
        Mean (expected value)<\/td>\n(a + b) \/ 2<\/strong><\/td>\n<\/tr>\n
        Variance<\/td>\n(b \u2212 a)\u00b2 \/ 12<\/strong><\/td>\n<\/tr>\n
        Standard deviation<\/td>\n(b \u2212 a) \/ \u221a12<\/strong><\/td>\n<\/tr>\n
        Probability density<\/td>\n1 \/ (b \u2212 a)<\/strong> inside interval<\/td>\n<\/tr>\n
        Cumulative distribution<\/td>\n(x \u2212 a) \/ (b \u2212 a)<\/strong> for a \u2264 x \u2264 b<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
        <\/div>\n<\/div>\n<\/div>\n
        Quick check:<\/p>\n
        \n
        \n
        \n
        Python<\/div>\n
        \n
        a, b = 15, 45\r\ndata = np.random.uniform(a, b, 100000)\r\n\r\nprint(\"Mean (should be ~30)    :\", data.mean().round(3))\r\nprint(\"Std  (should be ~8.66)  :\", data.std().round(3))\r\nprint(\"Theoretical std          :\", (b - a) \/ np.sqrt(12))<\/code><\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        7. Common student questions & confusions<\/h3>\n
        Q: Is uniform the same as random?<\/strong><\/p>\n
        No. Uniform means equal probability density<\/strong> in the interval. \u201cRandom\u201d is a much broader term \u2014 normal, binomial, exponential are also random, but not uniform.<\/p>\n
        Q: Why do we often see np.random.rand() instead of uniform(0,1)?<\/strong><\/p>\n
        rand() is just a convenient shortcut for uniform(0,1). They give exactly the same numbers (when seeded the same).<\/p>\n
        Q: Why does the histogram never look perfectly flat?<\/strong><\/p>\n
        Because it\u2019s a sample<\/strong>. With 100 values \u2192 looks bumpy. With 100,000 values \u2192 almost perfect rectangle.<\/p>\n
        Summary \u2013 Uniform Distribution Quick Reference<\/h3>\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>\nFlat rectangle<\/td>\n<\/tr>\n
        Defined by<\/td>\na (min), b (max)<\/td>\n<\/tr>\n
        Mean<\/td>\n(a + b)\/2<\/td>\n<\/tr>\n
        Variance<\/td>\n(b \u2212 a)\u00b2 \/ 12<\/td>\n<\/tr>\n
        NumPy standard [0,1)<\/td>\nnp.random.rand(size)<\/td>\n<\/tr>\n
        Custom interval<\/td>\nnp.random.uniform(a, b, size)<\/td>\n<\/tr>\n
        Most common use cases<\/td>\nmock data, initialization, test values, background noise, random sampling<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
        <\/div>\n<\/div>\n<\/div>\n
        Final teacher advice<\/h3>\n
        Whenever you need something \u201ccompletely fair \/ no preference \/ equal chance\u201d within a range<\/strong> \u2192 think uniform.<\/p>\n
        Whenever you see a flat histogram in real data<\/strong> (rare but happens in some sensors, quantization, etc.) \u2192 think uniform or discrete uniform.<\/p>\n
        Where would you like to go next?<\/p>\n
          \n
        • Difference between continuous vs discrete uniform<\/li>\n
        • How uniform is used in random number generation algorithms<\/li>\n
        • Transforming uniform to other distributions (inverse transform sampling)<\/li>\n
        • Comparing uniform vs normal vs exponential side by side<\/li>\n
        • Realistic mini-project: generate mock customer data + prices + timestamps<\/li>\n<\/ul>\n
          Just tell me what interests you most right now! \ud83d\ude0a<\/p>\n","protected":false},"excerpt":{"rendered":"
          1. What is the Uniform distribution really? The uniform distribution is the simplest and most intuitive continuous probability distribution. It means: Every value inside a certain interval is equally likely to occur. There are...<\/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-2512","post","type-post","status-publish","format-standard","hentry","category-numpy"],"_links":{"self":[{"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/posts\/2512","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=2512"}],"version-history":[{"count":1,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/posts\/2512\/revisions"}],"predecessor-version":[{"id":2513,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/posts\/2512\/revisions\/2513"}],"wp:attachment":[{"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/media?parent=2512"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/categories?post=2512"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/demo.materiamedica.net\/demo6\/wp-json\/wp\/v2\/tags?post=2512"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}