# Confidence interval for Poisson distribution

Let $y_1, y_2, \ldots,y_n$ be a simple random sample from a random variable $Y \sim Po(\lambda)$. I should calculate:

1. $\mathbb{E}(S_n)$ and $\mathbb{Var}(S_n)$, where $S_n=\sum_{i=1}^{n}Y_i$ and $Y_i$ are i.i.d.

2. An $1-\alpha$ confidence interval for $\tau=\frac{\mathbb{P}(Y_i=n)}{\mathbb{P}(Y_i=m)}$

Q1

I know that $\mathbb{E}(Y)=\lambda$ and $\mathbb{Var}(Y)=\lambda$, thus

$$\mathbb{E}(S_n)=n\lambda$$ and $$\mathbb{Var}(S_n)=n\lambda$$

Q2

Any suggest?

1. There are many ways to estimate $\tau$ as well as its standard error. The easiest and most cogent way of doing so is using the UMVUE estimate of $\lambda$, $\bar{Y} = S_n / n$. The probabilities in the numerator and denominator of $\tau$ can be expressed as a function of $\lambda$. Write them out exactly and find a simpler expression for $\tau$ as $f(\lambda$). Using that, an estimate of $\tau$ can be obtained using $\bar{Y}$ as a plugin estimate, its approximate distribution (according to the CLT), and the $\delta$-method to obtain standard errors for the approximate (limiting) normal distribution for your new estimator of $\tau$.
• HI, I started with $\tau=\frac{\mathbb{P}(Y_i=n)}{\mathbb{P}(Y_i=m)}=\frac{e^{-\lambda}\frac{\lambda^n}{n!}}{e^{-\lambda}\frac{\lambda^m}{m!}}=\frac{m!}{n!}\lambda^{n-m}$. Now I will try to follow your lead, thanks.
• @Paul : A known software bug sometimes prevents proper rendering of MathJax when the code has too many consecutive non-blank spaces. That's probably what happened in your comment above. $\qquad$ Jul 4 '16 at 17:24