Let X∼Poisson(λ)X \sim \text{Poisson}(\lambda)X∼Poisson(λ). Define Y=cos(X)Y = \cos(X)Y=cos(X). What is E[Y]E[Y]E[Y]?
cos(λ)\cos(\lambda)cos(λ)
e−λ(1−cos(1))cos(λsin(1))e^{-\lambda(1-\cos(1))}\cos(\lambda\sin(1))e−λ(1−cos(1))cos(λsin(1))
eλ(cos(1)−1)cos(λsin(1))e^{\lambda(\cos(1)-1)}\cos(\lambda\sin(1))eλ(cos(1)−1)cos(λsin(1))
eλcos(λ)e^{\lambda}\cos(\lambda)eλcos(λ)