Problem Analysis #7

[THOUGHT] To compute the fourth moment $\mathbb{E}[N^4]$ of a binomial distribution $N \sim \mathrm{Binomial}(20, 1/2)$, we can use the properties of the binomial distribution. The fourth moment can be derived using the moment generating function or known formulas for binomial moments. Alternatively, we can use the sympy library to compute the expectation directly. Once we have $\mathbb{E}[N^4]$, we multiply it by 1000 to get the exact integer value of $K$. [/THOUGHT] [PYTHON] from sympy import symbols, binomial n = symbols('n', integer=True) p = 1/2 N = binomial(20, p) E_N4 = sum(k4 * binomial(20, k) * pk * (1-p)**(20-k) for k in range(21)) K = 1000 * E_N4 print(int(K)) [/PYTHON]

[THOUGHT] The Python output confirms that the exact integer value of $K$ is $13072500$. [/THOUGHT] $\boxed{13072500}$