Data Science

A discrete distribution is a statistical distribution that shows the probabilities of outcomes as finite values
A discrete distribution is a function that maps some finite value from the sample space to the probability space
In other words, a discrete distribution represents the probabilities of discrete events across some pre-determined space

A bernoulli distribution is a discrete probability distribution of a random variable that takes the value 1 with a probability $p$ and the value 0 with probability $1-p$
We can define a bernoulli random variable as the following:

X \sim Bernoulli(p)

P(x) = p^{x}(1-p)^{1-x}

E(x) = p

Var(x) = p(1-p)

The range of a bernoulli distribution is $\{0,1\}$
A bernoulli distribution represents the probability of observing a success
In other words, a bernoulli distribution is represented by any two-element space
There is only one parameter that makes up the bernoulli distribution
Specifically, the parameter $p$ (or $\mu$ ) represents the probability of observing a 1
In other words, $P(X=1) = p$ or $P(X=1) = \mu$
The following are examples of random variables that are represented by a bernoulli distribution:
- A random variable that maps values to either heads or tails
- A random variable that maps values to either rain or shine
- A random variable that maps values to either democrat or republican
Bernoulli distributions are especially interesting because logistic regression assumes the observations of the response variable is a benoulli random variable

A binomial distribution is a discrete probability distribution of a random variable that represents the number of successes $p$ in a sample of size $n$
We can define a binomial random variable as the following:

X \sim Binomial(p;n)

P(x) = \binom{n}{k}p^{x}(1-p)^{n-x}

E(x) = np

Var(x) = np(1-p)

A binomial distribution represents a sequence of bernoulli trials (or a bernoulli process)
The range of a binomial distribution is $[0,1]$
A binomial distribution models the number of successes in a sample size (drawn with replacement from a population)
Therefore, there are two parameters that make up a binomial distribution
Specifically, the parameter $n$ represents the size of the sample and the parameter $p$ represents the probability of observing a success in the sample
Binomial distributions are especially interesting because logistic regression assumes the response variable represents a binomial random variable

A poisson distribution is a discrete probability distribution of a random variable that represents the number of successes occurring in a fixed interval of time
We can define a poisson random variable as the following:

X \sim Poisson(\lambda)

P(x) = \frac{\lambda^{k}e^{-\lambda}}{k!}

E(x) = \lambda \text{ where } \lim_{n \rarr \infty} \hat{\lambda} = \lambda \text{ where } \hat{\lambda} = \frac{1}{n}\sum_{i=1}^{n}X_i

Var(x) = \lambda \text{ where } \lim_{n \rarr \infty} \hat{\lambda} = \lambda \text{ where } \hat{\lambda} = \frac{1}{n}\sum_{i=1}^{n}X_i

A poisson distribution expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate $k$ (and independently of the time since the last event)
There is only one parameter that makes up a poisson distribution
Specifically, the parameter $\lambda$ represents the expected number of occurrences (and doesn't need to be an integer)
Poisson distributions are especially interesting because they are used in GLMs to model rates
Essentially, the poisson distribution is popular for modeling the number of times an event occurs in an interval of time or space
The following are examples of random variables that are represented by a poisson distribution:
- The number of meteorites greater than 1 meter diameter that strike Earth in a year
- The number of patients arriving in an emergency room between 10 and 11 pm
- The number of photons hitting a detector in a particular time interval

Overview of Probability

Continuous Distributions