Let X∼Poisson(λ)X \sim \text{Poisson}(\lambda)X∼Poisson(λ). What is the expected value of 2X2^X2X conditional on XXX being an even number?
cosh(2λ)cosh(λ)\frac{\cosh(2\lambda)}{\cosh(\lambda)}cosh(λ)cosh(2λ)
sinh(2λ)sinh(λ)\frac{\sinh(2\lambda)}{\sinh(\lambda)}sinh(λ)sinh(2λ)
eλe^{\lambda}eλ
e2λ+e−2λ2\frac{e^{2\lambda} + e^{-2\lambda}}{2}2e2λ+e−2λ