The number of requests at a Web server is a Poisson random...

The number of requests at a Web server is a Poisson random variable $X$ with mean $\alpha=2$ requests per minute. Suppose that $n 1$-minute intervals are observed and that the number $N_0$ of intervals with zero arrivals is counted. The probability of zero arrivals is then estimated by $\hat{p}_0=N_0 / n$. To estimate the arrival rate $\alpha, \hat{p}$ is set equal to the probability of zero arrivals in one minute: $$ \hat{p}_0=N_0 / n=P[X=0]=\frac{\alpha^0}{0!} e^{-\alpha}=e^{-\alpha} . $$ (a) Solve the above equation for $\hat{\alpha}$ to obtain an estimator for the arrival rate. (b) Show that $\hat{\alpha}$ is biased. (c) Find the mean square error of $\hat{\alpha}$. (d) Is $\hat{\alpha}$ a consistent estimator?

The number of requests at a Web server is a Poisson random variable $X$ with mean $\alpha=2$ requests per minute. Suppose that $n 1$-minute intervals are observed and that the number $N_0$ of intervals with zero arrivals is counted. The probability of zero arrivals is then estimated by $\hat{p}_0=N_0 / n$. To estimate the arrival rate $\alpha, \hat{p}$ is set equal to the probability of zero arrivals in one minute:

$$
\hat{p}_0=N_0 / n=P[X=0]=\frac{\alpha^0}{0!} e^{-\alpha}=e^{-\alpha} .
$$

(a) Solve the above equation for $\hat{\alpha}$ to obtain an estimator for the arrival rate.
(b) Show that $\hat{\alpha}$ is biased.
(c) Find the mean square error of $\hat{\alpha}$.
(d) Is $\hat{\alpha}$ a consistent estimator?

Key Concepts

Consistency of Estimators

Consistency refers to the property of an estimator whereby, as the sample size increases, the estimator converges in probability to the true parameter value. A consistent estimator is desirable because it ensures that with sufficient data, the estimation error can be made arbitrarily small. Investigating consistency involves examining the long?run behavior of the estimator as more data becomes available.

Mean Square Error (MSE)

The mean square error is a comprehensive measure used to evaluate the performance of an estimator. It is defined as the expected squared difference between the estimator and the true parameter, combining both the variance of the estimator and its squared bias. MSE provides a single metric that reflects the overall estimation error, making it a vital tool in assessing estimator efficiency.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that describes the number of events occurring in a fixed interval of time or space, given that these events happen at a constant mean rate independently of the time since the last event. It is widely used in scenarios where events occur randomly and independently, such as modeling counts in a web server context or other similar processes.

Parameter Estimation via Inversion

In many estimation problems, a known probability model is inverted to obtain an estimator for a parameter. In this context, a function of the parameter (for example, the probability of zero events) is equated to an empirical proportion, and solving this equation yields an estimator. This strategy demonstrates how transformation of variables can facilitate parameter estimation when dealing with nonlinear relationships.

Bias of Estimators

The bias of an estimator measures the difference between the expected value of the estimator and the true parameter value. An estimator is unbiased if its expected value equals the parameter, and biased if there is a systematic deviation. Bias analysis is crucial for understanding the properties of an estimator and for determining whether corrections or alternative methods are required to improve accuracy.

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