# How can you prove the Poisson Distribution?

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The Poisson distribution can be proven using the properties of the binomial distribution in the limit as the number of trials approaches infinity and the probability of success approaches zero, while the product of these two values remains constant. Specifically, as the number of trials (n) approaches infinity and the probability of success (p) approaches zero, with np being held constant, the binomial distribution converges to the Poisson distribution. This can be mathematically demonstrated through the limit definition and various probability theory concepts.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- The commuting time for all employees of a company has a normal distribution with a mean of 44.6 minutes and a standard deviation of 11.2 minutes. What is the probability that the mean computing time by a sample of 12 employees is between 43.26 and 49.35?
- How do you find the mean of the following set of numbers: 12, 14, 16, 12, 13?
- There are 8 members of the women’s basketball team and 7 members of the men’s track team at an athletic club meeting. What is the probability that a committee of 3 selected at random will have at least 2 members of the basketball team?
- Products from a certain machine are too large 15% of the time. What is the probability that in a run of 20 parts, 5 are too large?
- You keep track of the time you spend doing homework each evening. You spend 58 minutes, 36 minutes, 44 minutes, and 37 minutes. How do you find the mean of these times?

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