Mathematics, 30.07.2019 17:00 Porciabeauty4175
Let x be a random variable that takes integer values and is symmetric, that is, p(x = k) = p(x = âk) for all integers k. what is the expected value of y = cos(xï) and y = sin(xï)?
Answers: 1
is distributed, or what its PMF happens to be, we can't completely answer the question. However, recall that
, then![\mathbb E[Y]=\displaystyle\sum_{k\in\mathbb Z}\cos(k\pi)\,\mathbb P(X=k)](/tpl/images/0151/2483/0b422.png)
![\mathbb E[Y]=\mathbb P(X=0)+2\displaystyle\sum_{k\ge1}\cos(k\pi)\,\mathbb P(X=k)](/tpl/images/0151/2483/2ca73.png)
, and using the fact that 
for 
, we can write this as![\mathbb E[Y]=\mathbb P(X=0)+2\displaystyle\sum_{\ell\ge1}\bigg(\mathbb P(X=2\ell)-\mathbb P(X=2\ell-1)\bigg)](/tpl/images/0151/2483/d2f2e.png)
, we would have![\mathbb E[Y]=\mathbb E[\sin(\pi X)]=\displaystyle\sum_{k\in\mathbb Z}\sin(k\pi)\,\mathbb P(X=k)](/tpl/images/0151/2483/679b5.png)
for all integers ![\mathbb E[\sin(X\pi)]=0](/tpl/images/0151/2483/bf158.png)
.
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Let x be a random variable that takes integer values and is symmetric, that is, p(x = k) = p(x = âk)...
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