Let X1, X2, …, Xn denote a random sample from a Poisson...

Let $X_{1}, X_{2}, \ldots, X_{n}$ denote a random sample from a Poisson distribution with parameter $\theta>0$. From Remark 7.6.1, we know that $E\left[(-1)^{X_{1}}\right]=e^{-2 \theta}$. (a) Show that $E\left[(-1)^{X_{1}} \mid Y_{1}=y_{1}\right]=(1-2 / n)^{y_{1}}$, where $Y_{1}=X_{1}+X_{2}+\cdots+X_{n}$. Hint: First show that the conditional pdf of $X_{1}, X_{2}, \ldots, X_{n-1}$, given $Y_{1}=y_{1}$, is multinomial, and hence that of $X_{1}$, given $Y_{1}=y_{1}$, is $b\left(y_{1}, 1 / n\right)$. (b) Show that the mle of $e^{-2 \theta}$ is $e^{-2 \bar{X}}$. (c) Since $y_{1}=n \bar{x}$, show that $(1-2 / n)^{y_{1}}$ is approximately equal to $e^{-2 \pi}$ when $n$ is large.

Let $X_{1}, X_{2}, \ldots, X_{n}$ denote a random sample from a Poisson distribution with parameter $\theta>0$. From Remark 7.6.1, we know that $E\left[(-1)^{X_{1}}\right]=e^{-2 \theta}$.
(a) Show that $E\left[(-1)^{X_{1}} \mid Y_{1}=y_{1}\right]=(1-2 / n)^{y_{1}}$, where $Y_{1}=X_{1}+X_{2}+\cdots+X_{n}$.
Hint: First show that the conditional pdf of $X_{1}, X_{2}, \ldots, X_{n-1}$, given $Y_{1}=y_{1}$, is multinomial, and hence that of $X_{1}$, given $Y_{1}=y_{1}$, is $b\left(y_{1}, 1 / n\right)$.
(b) Show that the mle of $e^{-2 \theta}$ is $e^{-2 \bar{X}}$.
(c) Since $y_{1}=n \bar{x}$, show that $(1-2 / n)^{y_{1}}$ is approximately equal to $e^{-2 \pi}$ when $n$ is large.

Key Concepts

Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) is a method for estimating the parameters of a statistical model by maximizing the likelihood function, meaning one chooses the parameter values that make the observed data most probable. In this context, the MLE of the Poisson parameter using the sample mean is used to estimate functions of the parameter, such as e^(-2?), by directly plugging in the estimated value.

Asymptotic Approximation

Asymptotic approximation refers to methods used to approximate functions when parameters (such as the sample size) become large. A common example is the limit approximation (1 - a/n)^(bn) ? e^(-ab) as n grows large. This technique is crucial for simplifying expressions and understanding the behavior of estimators in large-sample settings.

Binomial Distribution

The binomial distribution, a special case of the multinomial distribution, arises when considering the count of successes in a fixed number of independent Bernoulli trials. In problems involving conditional distributions based on sums, the marginal distribution of a single count can be shown to be binomial, a key step in deriving moment conditions or conditional expectations.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space provided the events occur with a known constant mean rate and independently of the time since the last event. It is commonly used for modeling counting processes and has the important property that the sum of independent Poisson random variables is also Poisson distributed.

Conditional Distribution

The idea of a conditional distribution involves understanding the probability distribution of a subset of random variables given information about the sum or aggregate statistic, such as the total count. In contexts like this, the conditional distribution of individual counts given their sum can be derived and is instrumental in finding expectations and further properties.

Sum of Independent Poisson Random Variables

When you add independent Poisson random variables together, the result is also a Poisson random variable whose parameter is the sum of the individual parameters. This property is essential in simplifying worked problems involving several Poisson variables and is used to facilitate the derivation of the conditional distributions in such contexts.

Multinomial Distribution

The multinomial distribution generalizes the binomial distribution to experiments with more than two possible outcomes. In scenarios where the total number of events is fixed (for example, the sum of counts from several Poisson random variables), the joint distribution of counts across various categories, when conditioned on the total count, follows a multinomial distribution.

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