gistfile1.cpp

#include <iostream>
#include <iomanip>

// Expected value of an n-sided die.
double E_die(int n) {
    return (n + 1) / 2.;
}

// Variance of an n-sided die.
double V_die(int n) {
    double u = E_die(n);
    double p = 1. / n;

    double var = 0.;
    for (int i = 1; i <= n; ++i) {
        double t = (i - u);
        var += t*t*p;
    }

    return var;
}


int main(int argc, char** argv) {
    // The expectation of a random sum of N random variables:
    //
    //   |   N      |
    // E |  sum X_i |  = E[N]E[X]
    //   |  i=1     |
    //
    double E_T =        E_die(4);
    double E_C = E_T *  E_die(6);
    double E_O = E_C *  E_die(8);
    double E_D = E_O * E_die(12);
    double E_I = E_D * E_die(20);
    
    // The variance of a random sum of N random variables:
    //
    //     |   N      |
    // Var |  sum X_i |  = E[N] * Var[X] + E[X]^2 * Var[N]
    //     |  i=1     |
    //
    // So the variance of the sum of 1-4 6-sided die rolls is:
    // E_die(4) * V_die(6) + sq(E_die(6)) * V_die(4)
    double V_T =                              V_die(4);
    double V_C = V_T *  sq(E_die(6)) + E_T *  V_die(6);
    double V_O = V_C *  sq(E_die(8)) + E_C *  V_die(8);
    double V_D = V_O * sq(E_die(12)) + E_O * V_die(12);
    double V_I = V_D * sq(E_die(20)) + E_D * V_die(20);

    std::cout << std::fixed << std::setprecision(4) << V_I << "\n";

    return 0;
}