A student tosses a fair coin times and lets be the number of heads observed. Thus
For a complicated risk measure, the student needs the value of the fourth moment . Define
Compute the exact integer value of .
[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}