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Common Distributions

Here we will discuss some common distributions.

Binomial Distribution​

X∼B(n,p)X \sim B(n,p)

Introduction​

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials in an experiment with two possible outcomes, often labeled as "success" and "failure". Each trial, also known as a Bernoulli trial, has a constant probability of success, denoted by pp, and a constant probability of failure, 1−p1-p. The binomial distribution is defined by two parameters: the number of trials nn and the probability of success pp in each trial. The probability of getting exactly kk successes in nn trials is given by the formula (also called Probability Mass Function)

P(X=k)=(nk)pk(1−p)n−k,P(X=k)=\binom{n}{k}p^k(1-p)^{n-k},

where (nk)\binom{n}{k} is the binomial coefficient. This distribution is widely used in scenarios ranging from quality control in manufacturing to evaluating the probability of a certain number of successes in a set number of attempts in various fields such as biology, finance, and sports.

Expectation​

E[X]=∑k=0nkP(X=k)=∑k=0nk(nk)pk(1−p)n−k=∑k=1nk(nk)pk(1−p)n−k=∑k=1nkn!k!(n−k)!pk(1−p)n−k=np∑k=1n(n−1)!(k−1)!(n−k)!pk−1(1−p)n−k=np∑k=1n(n−1k−1)pk−1(1−p)n−k⏟PMF=np.\begin{align*} \mathbb{E}[X]&=\sum_{k=0}^n k P(X=k)\\ &=\sum_{k=0}^n k \binom{n}{k}p^k(1-p)^{n-k}\\ &=\sum_{k=1}^n k \binom{n}{k}p^k(1-p)^{n-k}\\ &=\sum_{k=1}^n k \frac{n!}{k!(n-k)!}p^k(1-p)^{n-k}\\ &=np\sum_{k=1}^n \frac{(n-1)!}{(k-1)!(n-k)!}p^{k-1}(1-p)^{n-k}\\ &=np\sum_{k=1}^n \underbrace{\binom{n-1}{k-1}p^{k-1}(1-p)^{n-k}}_{PMF}\\ &=np. \end{align*}

Variance​

Var[X]=\begin{align*} \mathbb{Var}[X]&= \end{align*}

Poisson Distribution​

X∼P(λ)X \sim P(\lambda)

Introduction​

The Poisson distribution is a discrete probability distribution that describes the probability of a given number of events occurring in a fixed interval of time or space if these events happen with a known constant rate and independently of the time since the last event. It is characterized by its lambda (λ)(\lambda) parameter, which represents the average number of events in the interval. The distribution is useful for modeling events with a low probability of occurrence over short intervals, such as the number of phone calls received by a call center in an hour or the number of decay events from a radioactive source in a day. The Poisson distribution assumes that these events are independent, meaning the occurrence of one event does not affect the probability of another event occurring in the same period. It is widely used in various fields such as physics, finance, and insurance, for modeling random events in time or space.