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A Poisson distribution is a probability distribution used to model the number of events occurring within a fixed interval of time or space. These events should be independent, and happening at a constant average rate. A key feature of this distribution is its mean \(\lambda\), which is equal to its variance. The mean \(\lambda\) represents both the expected number of events and the variability in these events. For instance, if a call center receives an average of 20 calls per hour, the distribution of the number of calls in an hour can be modeled with a Poisson distribution with \(\lambda = 20\). Understanding this concept is essential as it forms the basis in problems involving large samples that eventually relate to normal distributions.

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. This bell-shaped curve is important because it allows statisticians to assess probabilities and make inferences about sample data using the Z-scores. A Z-score indicates how many standard deviations an element is from the mean. In the original exercise, as the sample size grows, the Central Limit Theorem tells us that \(W_n\), a transformation of a Poisson-distributed variable, converges to a standard normal distribution. This convergence implies that we can use standard normal tables to find probabilities or critical values related to our sample statistic.

A confidence interval gives us a range of values which is likely to contain a population parameter, such as the mean \(\lambda\), with a specified level of confidence. In the context of the exercise, after showing the convergence of \(W_n\) to a standard normal distribution, we derive a formula to estimate \(\lambda\) with 95% confidence. The interval is constructed using the sample mean \(\bar{Y}\) and its standard error \(\sqrt{\bar{Y}/n}\). By calculating \(\bar{Y} \pm 1.96 \times \sqrt{\bar{Y}/n}\), we create a range in which \(\lambda\) is expected to fall 95% of the time, assuming the sample is large enough for the normal approximation to hold.

Slutsky's Theorem is a tool used in probability theory and statistics, which helps to justify approximations in many statistical problems, particularly when dealing with limits and convergence. The theorem states that if one sequence of random variables converges in distribution and another converges in probability to a constant, their combination also converges in distribution. In our exercise, it's used to solidify the conclusion that \(W_n\) converges to a standard normal distribution. By recognizing that \(\sqrt{n}(\bar{Y} - \lambda)/\sqrt{\lambda}\) approaches a normal distribution and \(\sqrt{\lambda}/\sqrt{\bar{Y}}\) converges to 1, Slutsky's theorem allows \(W_n\) to follow a normal distribution as well, bridging the conditions of convergence effectively.