#### Probability Distribution

- What is a probability distribution over a set S?
- Consider the set {*,+,-,:}. Construct a probability distribution over this set different in a way that all elements has distinct probabilities;
- Give the definition of the Cumulative Distribution Function of a random variable;
- Prove both Propositions in The Cumulative Distribution Function – MATH 4510/5510

#### Binomial Distribution

- Given a direct proof that a binomial distributed random variable X of parameters n and p has expect value equals np. (
*Hint: you can find a proof in the slides about binomial distribution. The important thing here is that you understand all the steps and know how to explain them*) - Prove the same result as above, but using that the expected value of the sum of random variables is the sum of the expected values.
*(Hint: Write X as sums of n variables taking values either 0 or 1;)*

#### Poisson Distribution

- Define the Poisson distribution of parameter
- What is the expected value of a Poisson distribution ?
- Describe how can a Poisson distribution be seen as a limit of Binomial distributions?
- Give an example of a concrete situation that can be modeled by a Poisson Distribution.
- Solve your own example. That is, give numbers to your situation and create a question which can answered using a Poisson Distribution

### Useful material

## Binomial Distribution – MATH 4510/5510

One of the most important discrete distributions

## Poisson Distribution – MATH 4510/5510

Introduction do the Poisson Distribution

## The Cumulative Distribution Function – MATH 4510/5510

The Cumulative Distribution Function and some properties