Probability Distributions

Binomial Distribution

  • The number of successes in a fixed number of independent trials, each with the same probability of success.
B(n,p)=∑k=0n(nk)pk(1−p)(n−k)=1B(n, p) = \sum^n_{k = 0} \begin{pmatrix}n \\ k\end{pmatrix}p^k(1 - p)^{(n -k)} = 1

∙\bullet Properties

  • Mean:
  • μ=np\mu = np
  • Variance:
  • σ2=np(1−p)\sigma^2 = np(1-p)
  • Probability mass function
  • P(X=k)=(nk)pk(1−p)(n−k)P(X = k) = \begin{pmatrix}n \\ k\end{pmatrix}p^k(1 - p)^{(n -k)}

Tip

Full proofs of properties can be found in Distribution Proofs

Poisson Distribution

P(λ)=∑k=0nk⋅λke−λk!P(\lambda) = \sum^{n}_{k = 0} k \cdot \frac{\lambda^k e^{-\lambda}}{k!}

∙\bullet Properties

  • Mean:
  • μ=λ\mu = \lambda
  • Variance:
  • σ2=λ\sigma^2 = \lambda
  • Probability mass function
  • P(X=k)=λke−λk!P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}