Random Sampling and Monte Carlo Intuition

/**
 * Fintech Math Bootcamp · Lesson 070 of 120
 * Random Sampling and Monte Carlo Intuition
 * Module 07: Probability Distributions and Simulation Basics
 *
 * Scenario: Estimating a quantity by simulation and checking the uncertainty
 * Rule:     Monte Carlo estimate = mean(g(Uᵢ))
 *
 * Try it:   Does increasing simulation count fix model misspecification?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/random-sampling-and-monte-carlo-intuition/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson070() {
  const draws=[.10,.40,.70,.80]; // frozen synthetic draw fixture
  const outputs=draws.map(u=>u*u);
  const estimate=outputs.reduce((s,x)=>s+x,0)/outputs.length;
  const result={outputs,estimate,exact:1/3,error:estimate-1/3};
  return result;
}

export const checkedResult = {"outputs":[0.010000000000000002,0.16000000000000003,0.48999999999999994,0.6400000000000001],"estimate":0.325,"exact":0.3333333333333333,"error":-0.008333333333333304};

// Run this file directly: npx tsx lessons/07-probability-distributions-and-simulation-basics/070-random-sampling-and-monte-carlo-intuition.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson070(), null, 2));
}

"""
Fintech Math Bootcamp · Lesson 070 of 120
Random Sampling and Monte Carlo Intuition
Module 07: Probability Distributions and Simulation Basics

Scenario: Estimating a quantity by simulation and checking the uncertainty
Rule:     Monte Carlo estimate = mean(g(Uᵢ))

Try it:   Does increasing simulation count fix model misspecification?

Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/random-sampling-and-monte-carlo-intuition/
Free course:    https://courses.thefintechbuilder.com
Synthetic teaching example, not financial advice or a production library.
"""

import json


def lesson_070():
    draws = [0.10, 0.40, 0.70, 0.80]  # frozen synthetic draw fixture
    outputs = [u * u for u in draws]
    total = 0
    for x in outputs:
        total += x
    estimate = total / len(outputs)
    return {"outputs": outputs, "estimate": estimate, "exact": 1 / 3, "error": estimate - 1 / 3}


if __name__ == "__main__":
    print(json.dumps(lesson_070(), indent=2))

// Fintech Math Bootcamp - Lesson 070 of 120
// Random Sampling and Monte Carlo Intuition
// Module 07: Probability Distributions and Simulation Basics
//
// Scenario: Estimating a quantity by simulation and checking the uncertainty
// Rule:     Monte Carlo estimate = mean(g(U_i))
//
// Try it:   Does increasing simulation count fix model misspecification?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/random-sampling-and-monte-carlo-intuition/
// 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> lesson070() {
        double[] draws = {0.10, 0.40, 0.70, 0.80}; // frozen synthetic draw fixture
        double[] outputs = new double[draws.length];
        double total = 0;
        for (int i = 0; i < draws.length; i++) {
            outputs[i] = draws[i] * draws[i];
            total += outputs[i];
        }
        double estimate = total / outputs.length;
        Map<String, Object> result = new LinkedHashMap<String, Object>();
        result.put("outputs", outputs);
        result.put("estimate", estimate);
        result.put("exact", 1.0 / 3);
        result.put("error", estimate - 1.0 / 3);
        return result;
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson070(), ""));
    }

    // 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 070 of 120
// Random Sampling and Monte Carlo Intuition
// Module 07: Probability Distributions and Simulation Basics
//
// Scenario: Estimating a quantity by simulation and checking the uncertainty
// Rule:     Monte Carlo estimate = mean(g(Uᵢ))
//
// Try it:   Does increasing simulation count fix model misspecification?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/random-sampling-and-monte-carlo-intuition/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
package main

import (
	"encoding/json"
	"fmt"
)

type Lesson070Result struct {
	Outputs  []float64 `json:"outputs"`
	Estimate float64   `json:"estimate"`
	Exact    float64   `json:"exact"`
	Error    float64   `json:"error"`
}

func lesson070() Lesson070Result {
	draws := []float64{0.10, 0.40, 0.70, 0.80} // frozen synthetic draw fixture
	outputs := make([]float64, len(draws))
	total := 0.0
	for i, u := range draws {
		outputs[i] = u * u
		total += outputs[i]
	}
	estimate := total / float64(len(outputs))
	return Lesson070Result{
		Outputs:  outputs,
		Estimate: estimate,
		Exact:    1.0 / 3,
		Error:    estimate - 1.0/3,
	}
}

func main() {
	out, err := json.MarshalIndent(lesson070(), "", "  ")
	if err != nil {
		panic(err)
	}
	fmt.Println(string(out))
}

// Fintech Math Bootcamp · Lesson 070 of 120
// Random Sampling and Monte Carlo Intuition
// Module 07: Probability Distributions and Simulation Basics
//
// Scenario: Estimating a quantity by simulation and checking the uncertainty
// Rule:     Monte Carlo estimate = mean(g(Uᵢ))
//
// Try it:   Does increasing simulation count fix model misspecification?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/random-sampling-and-monte-carlo-intuition/
// 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 lesson070() {
    const std::vector<double> draws = {0.10, 0.40, 0.70, 0.80};  // frozen synthetic draw fixture
    std::vector<double> outputs;
    double total = 0.0;
    for (double u : draws) {
        outputs.push_back(u * u);
        total += u * u;
    }
    const double estimate = total / outputs.size();
    return jsonObject({
        {"outputs", outputs},
        {"estimate", estimate},
        {"exact", 1.0 / 3},
        {"error", estimate - 1.0 / 3},
    });
}

int main() {
    std::cout << toJson(lesson070()) << '\n';
    return 0;
}

// Fintech Math Bootcamp · Lesson 070 of 120
// Random Sampling and Monte Carlo Intuition
// Module 07: Probability Distributions and Simulation Basics
//
// Scenario: Estimating a quantity by simulation and checking the uncertainty
// Rule:     Monte Carlo estimate = mean(g(Uᵢ))
//
// Try it:   Does increasing simulation count fix model misspecification?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/random-sampling-and-monte-carlo-intuition/
// 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_070() -> Json {
    let draws = [0.10_f64, 0.40, 0.70, 0.80]; // frozen synthetic draw fixture
    let outputs: Vec<f64> = draws.iter().map(|u| u * u).collect();
    let estimate = outputs.iter().fold(0.0_f64, |s, x| s + x) / outputs.len() as f64;
    Json::obj(vec![
        ("outputs", Json::nums(&outputs)),
        ("estimate", Json::Num(estimate)),
        ("exact", Json::Num(1.0 / 3.0)),
        ("error", Json::Num(estimate - 1.0 / 3.0)),
    ])
}

fn main() {
    println!("{}", lesson_070().pretty(""));
}

// Fintech Math Bootcamp · Lesson 070 of 120
// Random Sampling and Monte Carlo Intuition
// Module 07: Probability Distributions and Simulation Basics
//
// Scenario: Estimating a quantity by simulation and checking the uncertainty
// Rule:     Monte Carlo estimate = mean(g(Uᵢ))
//
// Try it:   Does increasing simulation count fix model misspecification?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-distributions-and-simulation-basics/random-sampling-and-monte-carlo-intuition/
// 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(Lesson070(), options));

static object Lesson070()
{
    double[] draws = { 0.10, 0.40, 0.70, 0.80 }; // frozen synthetic draw fixture
    double[] outputs = draws.Select(u => u * u).ToArray();
    double estimate = outputs.Aggregate(0.0, (s, x) => s + x) / outputs.Length;
    return new
    {
        outputs,
        estimate,
        exact = 1.0 / 3,
        error = estimate - 1.0 / 3,
    };
}