技术问询：设X为服从Binomial(n=4,p)的随机变量，求E(sin(nX/2))

Alright, let's work through this problem step by step. We have a random variable ( X \sim \text{Binomial}(n=4, p) ), and we need to calculate ( \mathbb{E}\left[\sin\left(\frac{nX}{2}\right)\right] ). First, since ( n=4 ), this simplifies to finding ( \mathbb{E}[\sin(2X)] ).

Key Background #

For any discrete random variable, the expectation of a function ( g(X) ) is the sum over all possible values of ( X ) of ( g(k) \cdot P(X=k) ). For our binomial distribution, the probability mass function is:
( P(X=k) = \binom{4}{k} p^k (1-p)^{4-k} )
where ( k = 0,1,2,3,4 ).

Calculating Each Term #

Let's compute the contribution of each possible ( k ) value to the expectation:

• When ( k=0 ): ( \sin(2*0) = \sin(0) = 0 ), so this term contributes 0.
• When ( k=1 ): ( \sin(2*1) = \sin(2) ), with ( P(X=1) = 4p(1-p)^3 ). Contribution: ( 4p(1-p)^3 \sin(2) ).
• When ( k=2 ): ( \sin(2*2) = \sin(4) ), with ( P(X=2) = 6p^2(1-p)2 ). Contribution: ( 6p^2(1-p)2 \sin(4) ).
• When ( k=3 ): ( \sin(2*3) = \sin(6) ), with ( P(X=3) = 4p^3(1-p) ). Contribution: ( 4p^3(1-p) \sin(6) ).
• When ( k=4 ): ( \sin(2*4) = \sin(8) ), with ( P(X=4) = p^4 ). Contribution: ( p^4 \sin(8) ).

Final Result #

Adding all these contributions together gives us the closed-form expectation:
( \mathbb{E}[\sin(2X)] = 4p(1-p)^3 \sin(2) + 6p^2(1-p)2 \sin(4) + 4p^3(1-p) \sin(6) + p^4 \sin(8) )

Alternative Verification with Characteristic Functions #

We can cross-verify this using the binomial characteristic function ( \phi_X(t) = \mathbb{E}[e^{itX}] = (pe^{it} + (1-p))^4 ). Using the identity ( \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i} ):
( \mathbb{E}[\sin(2X)] = \frac{1}{2i} \left( \phi_X(2) - \phi_X(-2) \right) )
Expanding ( (pe^{i2} + 1-p)^4 - (pe^{-i2} + 1-p)^4 ) and simplifying with ( e^{i\theta} - e^{-i\theta} = 2i\sin(\theta) ) will lead to the exact same result as above.

If you want to simplify the trigonometric terms (e.g., using ( \sin(6) = \sin(6-2\pi) = -\sin(2\pi-6) ) or angle addition formulas), the core expression remains valid as a complete solution.

内容的提问来源于stack exchange，提问作者hEcuLE