Hello every one,
I am doing a research paper, And I want to make sure that
if X is a sum of a*CosΘ
and Y is the Sum of a*SinΘ
my question, is the expected of E(sum(acosΘi))*E(sum(asinΘi))=0
where i = 1 to n
and for each Θ is independent uniform random variables on [0, 2π)
Thanks
Since Θ_i is the uniform random variable on [0, 2π] for each i = 1, 2,…,n, its probability density function f(x) is defined as f(x) = 1/(2π) if x ϵ [0, 2π] and f(x) = 0 if x < 0 or x > 2π. Hence, the expected value of the random variable cos Θ_i is equal to
E[cos Θ_i] = 1/(2π)ʃ_0^(2π) cos x dx = 0.
Hence, by the linearity of expected value of any random variable, we obtain that the expected value of the random variable X_n is equal to
E[X_n] = E[∑_(i = 1)^nacos Θ_i] = 0.
Proceeding in the same way with sin Θ_i instead of cos Θ_i, we find that that the expected value of the random variable Y_n is equal to
E[Y_n] = E[∑_(i = 1)^nasin Θ_i] = 0.
Hence, E[X_n] E[Y_n] = 0, as it was proposed in the question.
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