Let X∼Poisson(λ)X \sim \text{Poisson}(\lambda)X∼Poisson(λ). What is the conditional variance of XXX given that X≥1X \ge 1X≥1?
λ(1−(1+λ)e−λ)(1−e−λ)2\frac{\lambda(1 - (1+\lambda)e^{-\lambda})}{(1-e^{-\lambda})^2}(1−e−λ)2λ(1−(1+λ)e−λ)
λ1−e−λ\frac{\lambda}{1-e^{-\lambda}}1−e−λλ
λ−λ2e−λ(1−e−λ)2\lambda - \frac{\lambda^2 e^{-\lambda}}{(1-e^{-\lambda})^2}λ−(1−e−λ)2λ2e−λ
λ2+λ1−e−λ\frac{\lambda^2 + \lambda}{1-e^{-\lambda}}1−e−λλ2+λ