Explain

Intuition

Unbiased Estimator: When an estimator is unbiased, it means that if we could repeatedly sample data and compute the estimator, our guesses would "on average" be exactly right. We wouldn't consistently over-estimate or under-estimate the true parameter.

If we have one cell, the expected number of hits is $\lambda$. If we have 576 cells, the expected total hits is $576\lambda$. Dividing by 576, the average remains exactly $\lambda$. Because we just sum up the raw counts without distorting them and perfectly average them, our guess isn't artificially skewed higher or lower.

Estimator Variance: The term $\frac{\lambda}{N}$ tells us about the precision of our estimate.

Imagine you are trying to guess how many cars pass an intersection per minute.

• If you only count for $1$ minute (low $N$), your guess might be drastically wrong because car traffic fluctuates wildly.
• But if you count for $1000$ minutes (high $N$) and average the rate, your confidence in the average rate becomes much tighter and less prone to random flukes.

The variance $\frac{\lambda}{N}$ mathematically proves this: As your sample size $N$ (the number of grids or cells observed) grows larger, the variance of your estimate shrinks towards zero. With an infinitely large sample size, you would know the parameter $\lambda$ perfectly.

This relates to a fundamental concept in statistics known as the Law of Large Numbers.