Understanding the Unbiased Estimate for Poisson Parameter $\lambda$

The question asks for an unbiased estimate of the parameter $\lambda$ for a Poisson population, given a specific random sample. 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 if these events occur with a known constant mean rate and independently of the time since the last event.

The parameter $\lambda$ (lambda) represents the average rate of occurrence of events in the given interval. For a Poisson distribution, a key property is that its mean is equal to its variance, and both are equal to $\lambda$. That is, $\text{E}(X) = \lambda$ and $\text{Var}(X) = \lambda$ for a random variable $X$ following a Poisson distribution.

What is an Unbiased Estimator?

An estimator is a statistic used to estimate a population parameter. An estimator is said to be unbiased if its expected value is equal to the true value of the parameter being estimated. In simpler terms, on average, the estimator gets the value of the parameter right.

Finding the Unbiased Estimator for Poisson $\lambda$

For estimating the population mean ($\mu$) of any distribution, the sample mean ($\bar{X}$) is a commonly used estimator. A fundamental result in statistics is that the sample mean is an unbiased estimator for the population mean, i.e., $\text{E}(\bar{X}) = \mu$.

Since the mean of a Poisson distribution is equal to its parameter $\lambda$ (i.e., $\mu = \lambda$), the sample mean ($\bar{X}$) is an unbiased estimator for $\lambda$.

So, to find an unbiased estimate of $\lambda$ from the given sample, we need to calculate the sample mean.

Calculating the Sample Mean from the Given Data

The given random sample from the Poisson population is:

4, 5, 6, 6, 6, 6, 6, 6, 6, 7

The number of observations in the sample is $n=10$.

The sample mean, denoted by $\bar{X}$, is calculated as the sum of all observations divided by the number of observations:

$\bar{X} = \frac{\sum_{i=1}^{n} x_i}{n}$

First, let's find the sum of the observations:

$\sum x_i = 4 + 5 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 7$

$\sum x_i = 9 + 7 \times 6 + 7$

$\sum x_i = 9 + 42 + 7$

$\sum x_i = 58$

Now, divide the sum by the number of observations ($n=10$):

$\bar{X} = \frac{58}{10}$

$\bar{X} = 5.8$

Therefore, the sample mean is 5.8.

Conclusion: The Unbiased Estimate of $\lambda$

Since the sample mean ($\bar{X}$) is an unbiased estimator for the Poisson parameter $\lambda$, the unbiased estimate of $\lambda$ based on this sample is the calculated sample mean, which is 5.8.

The unbiased estimate of $\lambda$ is 5.8.

Revision Table: Key Statistical Concepts

Additional Information: Other Estimators and Properties

• Maximum Likelihood Estimator (MLE) for $\lambda$: For a Poisson distribution, the Maximum Likelihood Estimator for $\lambda$ is also the sample mean, $\bar{X}$. MLEs are often preferred due to properties like consistency and efficiency, although they are not always unbiased. In the case of the Poisson distribution, the sample mean is both an unbiased estimator and the MLE.
• Consistency: A consistent estimator is one that converges in probability to the true parameter value as the sample size increases. The sample mean is a consistent estimator for the population mean (and thus for $\lambda$).
• Efficiency: An efficient estimator is an unbiased estimator that has the smallest possible variance among all unbiased estimators. The sample mean is an efficient estimator for $\lambda$ in the Poisson distribution.
• Sufficiency: A sufficient statistic is one that captures all the information about the parameter that is available in the sample. The sample sum ($\sum x_i$) or equivalently the sample mean ($\bar{X}$) is a sufficient statistic for $\lambda$ in the Poisson distribution.