# Discrete Distributions

## Binomial Distribution

1. Bernoulli experiment
It has only two possible outcomes: success and failure. A success occurs with probability $p$, where $0<p<1$.

2. Binomial experiment
It is a Bernoulli experiment that is performed $n$ times, such that the different executions are performed independently of each other and with the same probability $p$.

3. Probability function
If a Bernoulli experiment with probability of success $p$ is performed $n$ times, and if $X$ denotes the total number of successes obtained, then the probability function of the binomial distribution with parameters $n$ and $p$ is given by:

$$f(k) = P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k=0,1,2, \dots, n$$

4. Cumulative distribution function
It is symbolized by $F$ and is defined as:

$$F(t) = P(X \leq t) = \sum_{k\leq t} f(k), \quad \text{for all real } t$$

5. Expected value and variance
If $X$ is binomial with parameters $n$ and $p$, then:

$$E(X)=np, \qquad V(X)= np(1-p)$$

## Poisson Distribution

1. Probability function
Let $X$ be the number of events occurring in a time interval $[0,t]$. The probability function of the Poisson distribution with parameter $\lambda >0$ is given by:

$$f(k) = P(X=k) = \frac{1}{k!} e^{-\lambda} \lambda^k, \quad k=0,1,2, 3, \dots$$

where $e$ is the base of the natural logarithm.

2. Cumulative distribution function
It is symbolized by $F$ and is defined as:

$$F(t) = P(X \leq t) = \sum_{k\leq t} f(k), \quad \text{for all real } t$$

3. Expected value and variance
If $X$ is Poisson with parameter $\lambda$, then:

$$E(X)= V(X)= \lambda$$

## Hypergeometric Distribution

1. Hypergeometric experiment
In general, a hypergeometric experiment with parameters $n$, $M$, and $N$ is based on the following assumptions:

- The population or set to be sampled is a finite population with $N$ elements.
- Each element of the population can be characterized as a success or a failure.
- There are $M$ successes in the population.
- A sample of $n$ individuals is chosen without replacement, such that every subset of size $n$ is equally likely to be selected.

2. Probability function
Let $X$ be the number of successes obtained in a randomly chosen sample when performing a hypergeometric experiment with parameters $n$, $M$, and $N$. Then, the probability of choosing exactly $k$ successes in $n$ trials is given by:

$$f(k) = P(X=k) = \frac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}}, \quad \text{where} \quad k=0,1,2, \dots, n \quad \text{and} \quad n\leq N$$

The corresponding distribution of $X$ is known as the hypergeometric distribution with parameters $n$, $M$, and $N$.

3. Cumulative distribution function
It is symbolized by $F$ and is defined as:

$$F(t) = P(X \leq t) = \sum_{k\leq t} f(k), \quad \text{for all real } t$$

4. Expected value and variance

$$E(X) = n\cdot\frac{M}{N} \qquad \text{and} \qquad V(X) = n \cdot\frac{M}{N} \cdot \left(1-\frac{M}{N}\right)\cdot \left(\frac{N-n}{N-1}\right)$$

## Negative Binomial Distribution

1. Negative binomial experiment
A negative binomial experiment with parameters $r$ and $p$ is characterized by the following conditions:

- The experiment consists of a series of Bernoulli experiments that are independent of each other.
- The probability of success $p$ for each Bernoulli experiment is always the same.
- The experiment continues until a total of $r$ successes have been observed, where $r$ is a given non-negative integer.

2. Probability function
Let $X$ be the number of failures preceding the $r$-th success in a negative binomial experiment with parameters $r$ and $p$. Then, the probability that there are $k$ failures before the $r$-th success is given by:

$$f(k) = P(X=k) = \binom{k+r-1}{r-1} p^r (1-p)^k, \qquad k=0,1,2, \dots$$

The corresponding distribution of $X$ is known as the negative binomial distribution with parameters $r$ and $p$.

3. Cumulative distribution function
It is symbolized by $F$ and is defined as:

$$F(t) = P(X \leq t) = \sum_{k\leq t} f(k), \quad \text{for all real } t$$

4. Expected value and variance

$$E(X) = \frac{r(1-p)}{p} \qquad \text{and} \qquad V(X) = \frac{r(1-p)}{p^2}$$

## Geometric Distribution

1. Special case
It is a special case of the negative binomial distribution with parameters $r=1$ and $p$.

2. Probability function
Let $X$ be the number of failures preceding the first success in a negative binomial experiment with parameters 1 and $p$. Then, the probability that there are $k$ failures before the first success is given by:

$$f(k) = P(X=k) = p (1-p)^k, \qquad k=0,1,2, \dots$$

The corresponding distribution of $X$ is known as the geometric distribution with parameter $p$.

3. Cumulative distribution function
It is symbolized by $F$ and is defined as:

$$F(t) = P(X \leq t) = \sum_{k\leq t} f(k), \quad \text{for all real } t$$

4. Expected value and variance

$$E(X)=\frac{1-p}{p}, \qquad V(X)=\frac{1-p}{p^2}$$