Covariance and Correlation of Random Variables

/**
 * Fintech Math Bootcamp · Lesson 060 of 120
 * Covariance and Correlation of Random Variables
 * Module 06: Probability and Random Variables
 *
 * Scenario: Dependence in shared credit or insurance losses
 * Rule:     Cov(X,Y)=E[(X−EX)(Y−EY)]
 *
 * Try it:   Are X and Y independent when Y=X² and covariance is zero?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/covariance-and-correlation-of-random-variables/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson060() {
  const states=[{x:-1,y:1,p:.25},{x:0,y:0,p:.5},{x:1,y:1,p:.25}];
  const ex=states.reduce((s,a)=>s+a.p*a.x,0);
  const ey=states.reduce((s,a)=>s+a.p*a.y,0);
  const covariance=states.reduce((s,a)=>s+a.p*(a.x-ex)*(a.y-ey),0);
  const result={ex,ey,covariance};
  return result;
}

export const checkedResult = {"ex":0,"ey":0.5,"covariance":0};

// Run this file directly: npx tsx lessons/06-probability-and-random-variables/060-covariance-and-correlation-of-random-variables.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson060(), null, 2));
}

# Fintech Math Bootcamp · Lesson 060 of 120
# Covariance and Correlation of Random Variables
# Module 06: Probability and Random Variables
#
# Scenario: Dependence in shared credit or insurance losses
# Rule:     Cov(X,Y)=E[(X−EX)(Y−EY)]
#
# Try it:   Are X and Y independent when Y=X² and covariance is zero?
#
# Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/covariance-and-correlation-of-random-variables/
# Free course:    https://courses.thefintechbuilder.com
# Synthetic teaching example, not financial advice or a production library.

import json


def lesson060() -> dict:
    states = [
        {"x": -1, "y": 1, "p": 0.25},
        {"x": 0, "y": 0, "p": 0.5},
        {"x": 1, "y": 1, "p": 0.25},
    ]
    ex = sum(s["p"] * s["x"] for s in states)
    ey = sum(s["p"] * s["y"] for s in states)
    covariance = sum(s["p"] * (s["x"] - ex) * (s["y"] - ey) for s in states)
    return {"ex": ex, "ey": ey, "covariance": covariance}


if __name__ == "__main__":
    print(json.dumps(lesson060(), indent=2))

/**
 * Fintech Math Bootcamp · Lesson 060 of 120
 * Covariance and Correlation of Random Variables
 * Module 06: Probability and Random Variables
 *
 * Scenario: Dependence in shared credit or insurance losses
 * Rule:     Cov(X,Y)=E[(X−EX)(Y−EY)]
 *
 * Try it:   Are X and Y independent when Y=X² and covariance is zero?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/covariance-and-correlation-of-random-variables/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

import java.util.ArrayList;
import java.util.LinkedHashMap;
import java.util.List;
import java.util.Map;

public class Main {
    // One joint outcome (x, y) with its probability p.
    static class State {
        final double x, y, p;

        State(double x, double y, double p) {
            this.x = x;
            this.y = y;
            this.p = p;
        }
    }

    static Map<String, Object> lesson060() {
        State[] states = {new State(-1, 1, 0.25), new State(0, 0, 0.5), new State(1, 1, 0.25)};
        double ex = 0, ey = 0;
        for (State s : states) {
            ex += s.p * s.x;
            ey += s.p * s.y;
        }
        double covariance = 0;
        for (State s : states) covariance += s.p * (s.x - ex) * (s.y - ey);
        Map<String, Object> result = new LinkedHashMap<String, Object>();
        result.put("ex", ex);
        result.put("ey", ey);
        result.put("covariance", covariance);
        return result;
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson060(), ""));
    }

    // --- Minimal JSON printer: maps keep insertion order, 2-space indent. ---

    static String toJson(Object value, String indent) {
        if (value == null) return "null";
        if (value instanceof String) return "\"" + value + "\"";
        if (value instanceof Double) return formatNumber((Double) value);
        if (value instanceof Number) return value.toString();
        if (value instanceof double[]) {
            List<Object> items = new ArrayList<Object>();
            for (double v : (double[]) value) items.add(v);
            return toJson(items, indent);
        }
        String inner = indent + "  ";
        StringBuilder sb = new StringBuilder();
        if (value instanceof Map) {
            Map<?, ?> map = (Map<?, ?>) value;
            if (map.isEmpty()) return "{}";
            sb.append("{\n");
            int i = 0;
            for (Map.Entry<?, ?> entry : map.entrySet()) {
                sb.append(inner).append('"').append(entry.getKey()).append("\": ")
                  .append(toJson(entry.getValue(), inner))
                  .append(++i < map.size() ? ",\n" : "\n");
            }
            return sb.append(indent).append('}').toString();
        }
        List<?> list = (List<?>) value;
        if (list.isEmpty()) return "[]";
        sb.append("[\n");
        for (int i = 0; i < list.size(); i++) {
            sb.append(inner).append(toJson(list.get(i), inner))
              .append(i + 1 < list.size() ? ",\n" : "\n");
        }
        return sb.append(indent).append(']').toString();
    }

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

// Fintech Math Bootcamp · Lesson 060 of 120
// Covariance and Correlation of Random Variables
// Module 06: Probability and Random Variables
//
// Scenario: Dependence in shared credit or insurance losses
// Rule:     Cov(X,Y)=E[(X−EX)(Y−EY)]
//
// Try it:   Are X and Y independent when Y=X² and covariance is zero?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/covariance-and-correlation-of-random-variables/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.

package main

import (
	"encoding/json"
	"fmt"
)

// State is one joint outcome (x, y) with its probability p.
type State struct {
	X, Y, P float64
}

// Covariance holds both expectations and the covariance of X and Y.
type Covariance struct {
	EX         float64 `json:"ex"`
	EY         float64 `json:"ey"`
	Covariance float64 `json:"covariance"`
}

func lesson060() Covariance {
	states := []State{{X: -1, Y: 1, P: 0.25}, {X: 0, Y: 0, P: 0.5}, {X: 1, Y: 1, P: 0.25}}
	ex, ey := 0.0, 0.0
	for _, s := range states {
		ex += s.P * s.X
		ey += s.P * s.Y
	}
	covariance := 0.0
	for _, s := range states {
		covariance += s.P * (s.X - ex) * (s.Y - ey)
	}
	return Covariance{EX: ex, EY: ey, Covariance: covariance}
}

func main() {
	out, _ := json.MarshalIndent(lesson060(), "", "  ")
	fmt.Println(string(out))
}

/**
 * Fintech Math Bootcamp · Lesson 060 of 120
 * Covariance and Correlation of Random Variables
 * Module 06: Probability and Random Variables
 *
 * Scenario: Dependence in shared credit or insurance losses
 * Rule:     Cov(X,Y)=E[(X−EX)(Y−EY)]
 *
 * Try it:   Are X and Y independent when Y=X² and covariance is zero?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/covariance-and-correlation-of-random-variables/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

#include <charconv>
#include <cmath>
#include <iostream>
#include <string>
#include <utility>
#include <vector>

// --- Minimal JSON value and printer: objects keep insertion order, 2-space indent. ---

struct Json {
    enum class Kind { Null, Number, Text, Array, Object };
    Kind kind = Kind::Null;
    double number = 0;
    std::string text;
    std::vector<std::string> keys;  // object keys, parallel to items
    std::vector<Json> items;        // array elements or object values
};

Json num(double v) { Json j; j.kind = Json::Kind::Number; j.number = v; return j; }
Json str(const std::string& s) { Json j; j.kind = Json::Kind::Text; j.text = s; return j; }
Json arr(const std::vector<Json>& values) { Json j; j.kind = Json::Kind::Array; j.items = values; return j; }

Json arr(const std::vector<double>& values) {
    std::vector<Json> items;
    for (double v : values) items.push_back(num(v));
    return arr(items);
}

Json obj(const std::vector<std::pair<std::string, Json>>& fields) {
    Json j;
    j.kind = Json::Kind::Object;
    for (const auto& [key, value] : fields) { j.keys.push_back(key); j.items.push_back(value); }
    return j;
}

std::string formatNumber(double v) {
    if (!std::isfinite(v)) return "null";
    char buf[64];
    auto end = std::to_chars(buf, buf + sizeof buf, v).ptr;  // shortest round-trip form
    return std::string(buf, end);
}

void writeJson(std::ostream& out, const Json& j, const std::string& indent) {
    switch (j.kind) {
    case Json::Kind::Null: out << "null"; return;
    case Json::Kind::Number: out << formatNumber(j.number); return;
    case Json::Kind::Text: out << '"' << j.text << '"'; return;
    default: break;
    }
    bool isObject = j.kind == Json::Kind::Object;
    if (j.items.empty()) { out << (isObject ? "{}" : "[]"); return; }
    std::string inner = indent + "  ";
    out << (isObject ? "{\n" : "[\n");
    for (size_t i = 0; i < j.items.size(); ++i) {
        out << inner;
        if (isObject) out << '"' << j.keys[i] << "\": ";
        writeJson(out, j.items[i], inner);
        out << (i + 1 < j.items.size() ? ",\n" : "\n");
    }
    out << indent << (isObject ? '}' : ']');
}

// --- Lesson ---

struct State {
    double x, y, p;  // one joint outcome (x, y) with its probability p
};

Json lesson060() {
    const std::vector<State> states = {{-1, 1, 0.25}, {0, 0, 0.5}, {1, 1, 0.25}};
    double ex = 0, ey = 0;
    for (const auto& s : states) {
        ex += s.p * s.x;
        ey += s.p * s.y;
    }
    double covariance = 0;
    for (const auto& s : states) covariance += s.p * (s.x - ex) * (s.y - ey);
    return obj({{"ex", num(ex)}, {"ey", num(ey)}, {"covariance", num(covariance)}});
}

int main() {
    writeJson(std::cout, lesson060(), "");
    std::cout << '\n';
}

// Fintech Math Bootcamp · Lesson 060 of 120
// Covariance and Correlation of Random Variables
// Module 06: Probability and Random Variables
//
// Scenario: Dependence in shared credit or insurance losses
// Rule:     Cov(X,Y)=E[(X−EX)(Y−EY)]
//
// Try it:   Are X and Y independent when Y=X² and covariance is zero?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/covariance-and-correlation-of-random-variables/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.

// --- Minimal JSON value and printer: objects keep insertion order, 2-space indent. ---

#[allow(dead_code)]
enum Json {
    Null,
    Num(f64),
    Str(String),
    Arr(Vec<Json>),
    Obj(Vec<(String, Json)>),
}

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

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

fn format_number(v: f64) -> String {
    if !v.is_finite() {
        return "null".to_string();
    }
    if v.fract() == 0.0 && v.abs() < 1e15 {
        return format!("{}", v as i64);
    }
    format!("{:?}", v) // shortest round-trip form
}

impl Json {
    fn render(&self, indent: &str) -> String {
        let inner = format!("{}  ", indent);
        match self {
            Json::Null => "null".to_string(),
            Json::Num(v) => format_number(*v),
            Json::Str(s) => format!("\"{}\"", s),
            Json::Arr(items) if items.is_empty() => "[]".to_string(),
            Json::Obj(fields) if fields.is_empty() => "{}".to_string(),
            Json::Arr(items) => {
                let lines: Vec<String> = items.iter().map(|v| format!("{}{}", inner, v.render(&inner))).collect();
                format!("[\n{}\n{}]", lines.join(",\n"), indent)
            }
            Json::Obj(fields) => {
                let lines: Vec<String> = fields
                    .iter()
                    .map(|(k, v)| format!("{}\"{}\": {}", inner, k, v.render(&inner)))
                    .collect();
                format!("{{\n{}\n{}}}", lines.join(",\n"), indent)
            }
        }
    }
}

// --- Lesson ---

/// One joint outcome (x, y) with its probability p.
struct State {
    x: f64,
    y: f64,
    p: f64,
}

fn lesson060() -> Json {
    let states = [
        State { x: -1.0, y: 1.0, p: 0.25 },
        State { x: 0.0, y: 0.0, p: 0.5 },
        State { x: 1.0, y: 1.0, p: 0.25 },
    ];
    let ex: f64 = states.iter().map(|s| s.p * s.x).sum();
    let ey: f64 = states.iter().map(|s| s.p * s.y).sum();
    let covariance: f64 = states.iter().map(|s| s.p * (s.x - ex) * (s.y - ey)).sum();
    obj(vec![("ex", Json::Num(ex)), ("ey", Json::Num(ey)), ("covariance", Json::Num(covariance))])
}

fn main() {
    println!("{}", lesson060().render(""));
}

/**
 * Fintech Math Bootcamp · Lesson 060 of 120
 * Covariance and Correlation of Random Variables
 * Module 06: Probability and Random Variables
 *
 * Scenario: Dependence in shared credit or insurance losses
 * Rule:     Cov(X,Y)=E[(X−EX)(Y−EY)]
 *
 * Try it:   Are X and Y independent when Y=X² and covariance is zero?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/probability-and-random-variables/covariance-and-correlation-of-random-variables/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

using System;
using System.Linq;
using System.Text.Json;

var options = new JsonSerializerOptions { WriteIndented = true };
Console.WriteLine(JsonSerializer.Serialize(Lesson060(), options));

static object Lesson060()
{
    var states = new[]
    {
        new { x = -1.0, y = 1.0, p = 0.25 },
        new { x = 0.0, y = 0.0, p = 0.5 },
        new { x = 1.0, y = 1.0, p = 0.25 },
    };
    double ex = states.Sum(s => s.p * s.x);
    double ey = states.Sum(s => s.p * s.y);
    double covariance = states.Sum(s => s.p * (s.x - ex) * (s.y - ey));
    return new { ex, ey, covariance };
}