Mixture Distributions, Multimodality, and Fat Tails

/**
 * Fintech Math Bootcamp · Lesson 069 of 120
 * Mixture Distributions, Multimodality, and Fat Tails
 * Module 07: Probability Distributions and Simulation Basics
 *
 * Scenario: Separating customer groups instead of forcing one average distribution
 * Rule:     mixture mean = Σwμ; variance = Σw[σ²+(μ−μmix)²]
 *
 * Try it:   Why is average component variance insufficient?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/mixture-distributions-multimodality-and-fat-tails/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson069() {
  const weights=[.8,.2], means=[1,5], variances=[1,1];
  const mean=weights.reduce((s,w,i)=>s+w*means[i],0);
  const variance=weights.reduce((s,w,i)=>s+w*(variances[i]+(means[i]-mean)**2),0);
  const result={mean,variance};
  return result;
}

export const checkedResult = {"mean":1.8,"variance":3.560000000000001};

// Run this file directly: npx tsx lessons/07-probability-distributions-and-simulation-basics/069-mixture-distributions-multimodality-and-fat-tails.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson069(), null, 2));
}

"""
Fintech Math Bootcamp · Lesson 069 of 120
Mixture Distributions, Multimodality, and Fat Tails
Module 07: Probability Distributions and Simulation Basics

Scenario: Separating customer groups instead of forcing one average distribution
Rule:     mixture mean = Σwμ; variance = Σw[σ²+(μ−μmix)²]

Try it:   Why is average component variance insufficient?

Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/mixture-distributions-multimodality-and-fat-tails/
Free course:    https://courses.thefintechbuilder.com
Synthetic teaching example, not financial advice or a production library.
"""

import json


def lesson_069():
    weights, means, variances = [0.8, 0.2], [1, 5], [1, 1]
    mean = 0
    for w, mu in zip(weights, means):
        mean += w * mu
    variance = 0
    for w, mu, var in zip(weights, means, variances):
        variance += w * (var + (mu - mean) ** 2)
    return {"mean": mean, "variance": variance}


if __name__ == "__main__":
    print(json.dumps(lesson_069(), indent=2))

// Fintech Math Bootcamp - Lesson 069 of 120
// Mixture Distributions, Multimodality, and Fat Tails
// Module 07: Probability Distributions and Simulation Basics
//
// Scenario: Separating customer groups instead of forcing one average distribution
// Rule:     mixture mean = sum(w*mu); variance = sum(w*[sigma^2+(mu-mu_mix)^2])
//
// Try it:   Why is average component variance insufficient?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/mixture-distributions-multimodality-and-fat-tails/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
import java.util.ArrayList;
import java.util.Arrays;
import java.util.LinkedHashMap;
import java.util.List;
import java.util.Map;

public class Main {
    static Map<String, Object> lesson069() {
        double[] weights = {0.8, 0.2}, means = {1, 5}, variances = {1, 1};
        double mean = 0;
        for (int i = 0; i < weights.length; i++) {
            mean += weights[i] * means[i];
        }
        double variance = 0;
        for (int i = 0; i < weights.length; i++) {
            variance += weights[i] * (variances[i] + Math.pow(means[i] - mean, 2));
        }
        Map<String, Object> result = new LinkedHashMap<String, Object>();
        result.put("mean", mean);
        result.put("variance", variance);
        return result;
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson069(), ""));
    }

    // Minimal JSON writer: two-space indent, whole numbers without a decimal point, NaN as null.
    static String toJson(Object value, String indent) {
        if (value == null) return "null";
        if (value instanceof Boolean) return value.toString();
        if (value instanceof Number) return formatNumber(((Number) value).doubleValue());
        if (value instanceof String) return quote((String) value);
        if (value instanceof double[]) {
            List<Object> boxed = new ArrayList<Object>();
            for (double d : (double[]) value) boxed.add(d);
            return toJson(boxed, indent);
        }
        if (value instanceof Object[]) return toJson(Arrays.asList((Object[]) value), indent);
        String inner = indent + "  ";
        StringBuilder out = new StringBuilder();
        if (value instanceof Map) {
            Map<?, ?> map = (Map<?, ?>) value;
            if (map.isEmpty()) return "{}";
            out.append("{\n");
            int i = 0;
            for (Map.Entry<?, ?> entry : map.entrySet()) {
                out.append(inner).append(quote(entry.getKey().toString())).append(": ")
                   .append(toJson(entry.getValue(), inner));
                out.append(++i < map.size() ? ",\n" : "\n");
            }
            return out.append(indent).append("}").toString();
        }
        List<?> list = (List<?>) value;
        if (list.isEmpty()) return "[]";
        out.append("[\n");
        for (int i = 0; i < list.size(); i++) {
            out.append(inner).append(toJson(list.get(i), inner));
            out.append(i + 1 < list.size() ? ",\n" : "\n");
        }
        return out.append(indent).append("]").toString();
    }

    static String formatNumber(double x) {
        if (Double.isNaN(x) || Double.isInfinite(x)) return "null";
        if (x == Math.rint(x) && Math.abs(x) < 1e15) return Long.toString((long) x);
        return Double.toString(x);
    }

    static String quote(String s) {
        StringBuilder out = new StringBuilder("\"");
        for (char c : s.toCharArray()) {
            if (c == '"' || c == '\\') out.append('\\').append(c);
            else if (c == '\n') out.append("\\n");
            else if (c < 0x20) out.append(String.format("\\u%04x", (int) c));
            else out.append(c);
        }
        return out.append('"').toString();
    }
}

// Fintech Math Bootcamp · Lesson 069 of 120
// Mixture Distributions, Multimodality, and Fat Tails
// Module 07: Probability Distributions and Simulation Basics
//
// Scenario: Separating customer groups instead of forcing one average distribution
// Rule:     mixture mean = Σwμ; variance = Σw[σ²+(μ−μmix)²]
//
// Try it:   Why is average component variance insufficient?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/mixture-distributions-multimodality-and-fat-tails/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
package main

import (
	"encoding/json"
	"fmt"
	"math"
)

type Lesson069Result struct {
	Mean     float64 `json:"mean"`
	Variance float64 `json:"variance"`
}

func lesson069() Lesson069Result {
	weights := []float64{0.8, 0.2}
	means := []float64{1, 5}
	variances := []float64{1, 1}
	mean := 0.0
	for i, w := range weights {
		mean += w * means[i]
	}
	variance := 0.0
	for i, w := range weights {
		variance += w * (variances[i] + math.Pow(means[i]-mean, 2))
	}
	return Lesson069Result{Mean: mean, Variance: variance}
}

func main() {
	out, err := json.MarshalIndent(lesson069(), "", "  ")
	if err != nil {
		panic(err)
	}
	fmt.Println(string(out))
}

// Fintech Math Bootcamp · Lesson 069 of 120
// Mixture Distributions, Multimodality, and Fat Tails
// Module 07: Probability Distributions and Simulation Basics
//
// Scenario: Separating customer groups instead of forcing one average distribution
// Rule:     mixture mean = Σwμ; variance = Σw[σ²+(μ−μmix)²]
//
// Try it:   Why is average component variance insufficient?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/mixture-distributions-multimodality-and-fat-tails/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <iostream>
#include <optional>
#include <stdexcept>
#include <string>
#include <utility>
#include <vector>

// A minimal JSON value, enough to print this lesson's result.
struct Json {
    enum class Kind { Null, Bool, Number, String, Array, Object };
    Kind kind = Kind::Null;
    bool flag = false;
    double number = 0.0;
    std::string text;
    std::vector<std::string> keys;  // object keys, parallel to items
    std::vector<Json> items;        // array elements or object values

    Json() = default;
    Json(bool value) : kind(Kind::Bool), flag(value) {}
    Json(int value) : kind(Kind::Number), number(value) {}
    Json(double value) : kind(Kind::Number), number(value) {}
    Json(const char* value) : kind(Kind::String), text(value) {}
    Json(const std::string& value) : kind(Kind::String), text(value) {}
    Json(const std::vector<double>& values) : kind(Kind::Array) {
        for (double v : values) items.push_back(Json(v));
    }
};

Json jsonArray(const std::vector<Json>& values) {
    Json array;
    array.kind = Json::Kind::Array;
    array.items = values;
    return array;
}

Json jsonObject(const std::vector<std::pair<std::string, Json>>& fields) {
    Json object;
    object.kind = Json::Kind::Object;
    for (const auto& field : fields) {
        object.keys.push_back(field.first);
        object.items.push_back(field.second);
    }
    return object;
}

// Shortest decimal form that reads back as the same double.
std::string formatNumber(double x) {
    if (!std::isfinite(x)) return "null";
    char buffer[32];
    if (x == std::floor(x) && std::fabs(x) < 1e15) {
        std::snprintf(buffer, sizeof buffer, "%.0f", x);
        return buffer;
    }
    for (int precision = 1; precision <= 17; ++precision) {
        std::snprintf(buffer, sizeof buffer, "%.*g", precision, x);
        if (std::strtod(buffer, nullptr) == x) break;
    }
    return buffer;
}

std::string quote(const std::string& s) {
    std::string out = "\"";
    for (char c : s) {
        if (c == '"' || c == '\\') { out += '\\'; out += c; }
        else if (c == '\n') out += "\\n";
        else out += c;
    }
    return out + "\"";
}

std::string toJson(const Json& value, const std::string& indent = "") {
    switch (value.kind) {
        case Json::Kind::Null: return "null";
        case Json::Kind::Bool: return value.flag ? "true" : "false";
        case Json::Kind::Number: return formatNumber(value.number);
        case Json::Kind::String: return quote(value.text);
        default: break;
    }
    const bool isObject = value.kind == Json::Kind::Object;
    if (value.items.empty()) return isObject ? "{}" : "[]";
    const std::string inner = indent + "  ";
    std::string out = isObject ? "{\n" : "[\n";
    for (std::size_t i = 0; i < value.items.size(); ++i) {
        out += inner;
        if (isObject) out += quote(value.keys[i]) + ": ";
        out += toJson(value.items[i], inner);
        out += i + 1 < value.items.size() ? ",\n" : "\n";
    }
    return out + indent + (isObject ? "}" : "]");
}

Json lesson069() {
    const std::vector<double> weights = {0.8, 0.2}, means = {1, 5}, variances = {1, 1};
    double mean = 0.0;
    for (std::size_t i = 0; i < weights.size(); ++i) mean += weights[i] * means[i];
    double variance = 0.0;
    for (std::size_t i = 0; i < weights.size(); ++i) {
        variance += weights[i] * (variances[i] + std::pow(means[i] - mean, 2));
    }
    return jsonObject({
        {"mean", mean},
        {"variance", variance},
    });
}

int main() {
    std::cout << toJson(lesson069()) << '\n';
    return 0;
}

// Fintech Math Bootcamp · Lesson 069 of 120
// Mixture Distributions, Multimodality, and Fat Tails
// Module 07: Probability Distributions and Simulation Basics
//
// Scenario: Separating customer groups instead of forcing one average distribution
// Rule:     mixture mean = Σwμ; variance = Σw[σ²+(μ−μmix)²]
//
// Try it:   Why is average component variance insufficient?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/mixture-distributions-multimodality-and-fat-tails/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.

/// A minimal JSON value, enough to print this lesson's result.
#[allow(dead_code)]
enum Json {
    Null,
    Bool(bool),
    Num(f64),
    Str(String),
    Arr(Vec<Json>),
    Obj(Vec<(String, Json)>),
}

#[allow(dead_code)]
impl Json {
    fn obj(fields: Vec<(&str, Json)>) -> Json {
        Json::Obj(fields.into_iter().map(|(k, v)| (k.to_string(), v)).collect())
    }

    fn nums(values: &[f64]) -> Json {
        Json::Arr(values.iter().map(|&v| Json::Num(v)).collect())
    }

    /// Pretty-prints with two-space indentation.
    fn pretty(&self, indent: &str) -> String {
        let inner = format!("{}  ", indent);
        match self {
            Json::Null => "null".to_string(),
            Json::Bool(b) => b.to_string(),
            Json::Num(x) => format_number(*x),
            Json::Str(s) => quote(s),
            Json::Arr(items) if items.is_empty() => "[]".to_string(),
            Json::Obj(fields) if fields.is_empty() => "{}".to_string(),
            Json::Arr(items) => {
                let body: Vec<String> = items
                    .iter()
                    .map(|v| format!("{}{}", inner, v.pretty(&inner)))
                    .collect();
                format!("[\n{}\n{}]", body.join(",\n"), indent)
            }
            Json::Obj(fields) => {
                let body: Vec<String> = fields
                    .iter()
                    .map(|(k, v)| format!("{}{}: {}", inner, quote(k), v.pretty(&inner)))
                    .collect();
                format!("{{\n{}\n{}}}", body.join(",\n"), indent)
            }
        }
    }
}

fn format_number(x: f64) -> String {
    if !x.is_finite() {
        "null".to_string()
    } else if x == x.trunc() && x.abs() < 1e15 {
        format!("{}", x as i64)
    } else {
        format!("{}", x)
    }
}

fn quote(s: &str) -> String {
    let mut out = String::from("\"");
    for c in s.chars() {
        match c {
            '"' => out.push_str("\\\""),
            '\\' => out.push_str("\\\\"),
            '\n' => out.push_str("\\n"),
            c => out.push(c),
        }
    }
    out.push('"');
    out
}

fn lesson_069() -> Json {
    let weights = [0.8_f64, 0.2];
    let means = [1.0_f64, 5.0];
    let variances = [1.0_f64, 1.0];
    let mean = weights
        .iter()
        .zip(means.iter())
        .fold(0.0_f64, |s, (w, mu)| s + w * mu);
    let variance = (0..weights.len()).fold(0.0_f64, |s, i| {
        s + weights[i] * (variances[i] + (means[i] - mean).powf(2.0))
    });
    Json::obj(vec![
        ("mean", Json::Num(mean)),
        ("variance", Json::Num(variance)),
    ])
}

fn main() {
    println!("{}", lesson_069().pretty(""));
}

// Fintech Math Bootcamp · Lesson 069 of 120
// Mixture Distributions, Multimodality, and Fat Tails
// Module 07: Probability Distributions and Simulation Basics
//
// Scenario: Separating customer groups instead of forcing one average distribution
// Rule:     mixture mean = Σwμ; variance = Σw[σ²+(μ−μmix)²]
//
// Try it:   Why is average component variance insufficient?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/mixture-distributions-multimodality-and-fat-tails/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text.Json;

var options = new JsonSerializerOptions { WriteIndented = true };
Console.WriteLine(JsonSerializer.Serialize(Lesson069(), options));

static object Lesson069()
{
    double[] weights = { 0.8, 0.2 }, means = { 1, 5 }, variances = { 1, 1 };
    double mean = weights.Select((w, i) => w * means[i]).Aggregate(0.0, (s, x) => s + x);
    double variance = weights
        .Select((w, i) => w * (variances[i] + Math.Pow(means[i] - mean, 2)))
        .Aggregate(0.0, (s, x) => s + x);
    return new { mean, variance };
}