Poisson distribution is a 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 average rate and independently of the time since the last event.

The Normal distribution is a continuous probability distribution that is symmetrical and bell-shaped, with a mean and standard deviation that determine its shape.

Poisson distribution approaches a Normal distribution as n (the sample size) increases infinitely. This is known as the Poisson approximation to the Normal distribution.

Explanation:

The Poisson distribution has some properties that make it useful for modeling the occurrence of rare events. However, as the sample size increases, the Poisson distribution becomes increasingly similar to the Normal distribution.

The Poisson distribution has a mean equal to its parameter λ, which represents the average rate of events occurring in the given interval. It also has a variance equal to its mean, which is λ. As λ increases, the shape of the Poisson distribution becomes more and more like a Normal distribution.

When the sample size is large, the Central Limit Theorem states that the sample means will be approximately normally distributed, regardless of the underlying distribution of the population. In the case of Poisson distribution, as λ increases, the sample size increases, and the sample means become more and more normally distributed.

The Normal distribution has several properties that make it useful for statistical analysis, such as the ability to calculate probabilities and confidence intervals. Therefore, the Poisson approximation to the Normal distribution is useful in situations where the sample size is large, and the Poisson distribution is difficult to work with.

Conclusion:

In summary, the Poisson distribution approaches a Normal distribution as the sample size increases infinitely. This is because the Poisson distribution becomes more and more like a Normal distribution as the average rate of events occurring in the given interval increases. The Poisson approximation to the Normal distribution is useful in situations where the sample size is large, and the Poisson distribution is difficult to work with.