Under which of the following conditions the Poisson distribution is the limiting case of, binomial distribution:

(A) The number of trials is indefinitely large.

(B) The probability of success for each trial is indefinitely small.

(C) The product of the number of trials and the probability of success for each trial is finite.

(D) The probability of success for each trial is indefinitely large.

Choose the correct answer from the options given below:

The Poisson distribution arises as a limiting case of the binomial distribution under specific conditions.

For a binomial distribution with $n$ trials and probability of success $p$, the transformation to Poisson occurs when three conditions are met simultaneously.

Condition (A): The number of trials is indefinitely large.

This requires $n \to \infty$.

When there are many opportunities for an event to occur, the framework shifts toward counting rare events over many trials, which is characteristic of the Poisson distribution.

Condition (B): The probability of success for each trial is indefinitely small.

This requires $p \to 0$.

Each individual trial must have a very small chance of success. The event must be rare on any single trial.

Condition (C): The product of the number of trials and the probability of success is finite.

This requires $np = \lambda$ where $\lambda$ is a finite constant.

Even though $n$ is very large and $p$ is very small, their product must remain constant. This product becomes the rate parameter of the Poisson distribution. If $np \to \infty$ or $np \to 0$, the Poisson distribution does not emerge.

Condition (D): The probability of success for each trial is indefinitely large.

This suggests $p \to 1$ or $p$ becoming large.

This contradicts condition (B), which requires $p \to 0$. A large probability of success does not lead to the Poisson limiting case.

The Poisson distribution is the limiting case of the binomial distribution when conditions (A), (B), and (C) hold simultaneously.

The correct answer is the option that includes (A), (B), and (C) only.