Let $X$ be a random variable with Binomial$(n = 4, p)$. Find $E (\sin \frac{nX}{2})$.

Question

Let $X$ be a random variable with $Binomial(n = 4, p)$. Find $E (\sin \frac{nX}{2})$ where $E(X)$ means expectation of $X$

Answer 1

Let's start:$$E(\sin\dfrac{nX}{2})=E(\sin\dfrac{4X}{2})=E(\sin 2X)=\int_{-\infty}^{\infty}\sin2xf_X(x)dx$$also $$P(X=k)=f_k=\binom{4}{k}p^k(1-p)^{4-k}$$therefore$$f_X(x)=f_0\delta(x)+f_1\delta(x-1)+f_2\delta(x-2)+f_3\delta(x-3)+f_4\delta(x-4)$$substituting this in the expression of the expectation leads us to:$$\int_{-\infty}^{\infty}\sin2xf_X(x)dx=f_0\sin0+f_1\sin1+f_2\sin2+f_3\sin3+f_4\sin4\\=f_1\sin1+f_2\sin2+f_3\sin3+f_4\sin4$$