Prediction of a Future Value of a Random Variable

Question

Suppose that $X∼\mathrm{Poisson}(5).$ What value would you record as a prediction of a future value of $X?$ How would you justify your choice?

Answer 1

Predict with what accuracy? If you want an approximately 95% prediction set, then maybe $x\in [1,9].$ In R:

qpois(c(.025,.975), 5)
[1]  1 10
sum(dpois(1:9, 5))
[1] 0.961434

x = 0:15;  pdf = dpois(x, 5)
plot(x, pdf, type="h", lwd=3, 
     ylab="PDF", main="POIS(5)")
 abline(v = c(.5, 9.5), col="red", lty="dotted")
 abline(h=0, col="green2");  abline(v=0, col="green2")

Simulation:

set.seed(2020)
x = rpois(10^6, 5)
mean(x >= 1 & x <=9)
[1] 0.961627

Note: Although a normal approximation to $\mathsf{Pois}(\lambda=5)$ is not very good, we have $\mu=\sigma^2 = 5, \sigma = \sqrt{5},$ so $\mu \pm 1.96\sigma$ gives the same set of integers. [This method works quite reliably for larger $\lambda.$ For example, if $Y\sim \mathsf{Pois}(\lambda = 25),$ then $P(16 \le Y \le 34) \approx 0.94.]$