Continuation distributions

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Created by
Merel DJ

Cards (8)

  • A function p : R -> R is a Probability Density Function (PDF) for a continuous variable X if
    p(x)≥∀x∈Val(x)p(x) \geq \forall x \in Val(x)
  • A function p : R -> R is a Probability Density Function (PDF) for a continuous variable X if :
    • p(x)≥∀x∈Val(X)p(x) \geq \forall x \in Val(X)(x) \geq \forall x \in Val(X)
    • p(x)=p(x) =0,∀x∉Val(X) 0, \forall x \notin Val(X)(x) = 0, \forall x \notin Val(X)
    • ∫−inf⁡+inf⁡p(x)dx=\int_{- \inf}^{+ \inf} p(x) dx =1 1
  • PDF is not a probability distribution
  • Cumultative Distribution Function (CDF)
    for any a in R : P(X≤a)=P(X \leq a) =∫−∞ap(x)dx \int_{- \infin}^{a} p(x) dx
  • P(a≤X≤b)=P(a \leq X \leq b) =P(X≤b)−P(X≤a)= P(X \leq b) - P(X \leq a) =∫abp(x)dx \int_a^b p(x)dx
  • Joint (Multivariate) Normal Distribution
    mean vector u, covariance sigma, denoted X ~ N(u, sigma) if they have the multivariate Gaussian PDF : p(x)=p(x) =1(2π)N/2∣Σ∣12−12(x−μ)TΣ−1(x−μ) \frac{1}{(2\pi)^{N/2} |\Sigma|^\frac{1}{2}}^{-\frac{1}{2}(x- \mu)^T \Sigma^{-1} (x - \mu)}
  • A variable X has a Normal Distirbution with mean mu and variable sigma^2 denoted X~{mean, variance^2}, p(x)=p(x) =12πσe−(x−μ)22σ2 \frac{1}{\sqrt{2\pi} \sigma}e^{- \frac{(x - \mu)^2}{2 \sigma^2}}
  • A real variable X has a Uniform distribution over range [a,b] denoted X ~ Unif[a,b], if it has the PDF :