# Study Guide: Probability Distributions

#### Probability Distribution

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

#### Binomial Distribution

1. 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 )
2. 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

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