A student tosses a fair coin times and lets be the number of heads observed. Thus
\nFor a complicated risk measure, the student needs the value of the fourth moment . \nDefine
\nCompute the exact integer value of .
[THOUGHT] I need to compute the fourth moment of a binomial random variable N ~ Binomial(20, 1/2). The fourth moment is E[N^4], and then K is defined as 1000 times that value. Since the problem requires an exact integer, I'll calculate the sum of k^4 multiplied by the binomial coefficient C(20, k) for k from 0 to 20, then multiply by 1000 and divide by 2^20 (which is 1048576). The result must be an integer as specified.
[PYTHON] import math total = 0 for k in range(0, 21): c = math.comb(20, k) term = k**4 * c total += term K = (1000 * total) // 1048576 print(K) [/PYTHON]
[THOUGHT]
The Python code computed the fourth moment of a binomial random variable N ~ Binomial(20, 1/2) by summing k^4 * C(20, k) for k=0 to 20, then multiplied by 1000 and divided by 2^20. The output 13072500 is the exact integer value of K.
[/THOUGHT]