Continuation distributions

• A function p : R -> R is a Probability Density Function (PDF) for a continuous variable X if

  $p(x) \geq \forall x \in Val(x)$
• A function p : R -> R is a Probability Density Function (PDF) for a continuous variable X if :

  • $p(x) \geq \forall x \in Val(X)$(x) \geq \forall x \in Val(X)
  • $p(x) =$$0, \forall x \notin Val(X)$(x) = 0, \forall x \notin Val(X)
  • $\int_{- \inf}^{+ \inf} p(x) dx =$$1$
• PDF is not a probability distribution
• Cumultative Distribution Function (CDF)

  for any a in R : $P(X \leq a) =$$\int_{- \infin}^{a} p(x) dx$
• $P(a \leq X \leq b) =$$P(X \leq b) - P(X \leq a) =$$\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) =$$\frac{1}{(2\pi)^{N/2} |\Sigma|^\frac{1}{2}}^{-\frac{1}{2}(x- \mu)^T \Sigma^{-1} (x - \mu)}$
• A variable X has a Normal Distirbution with mean mu and variable sigma^2 denoted X~{mean, variance^2}, $p(x) =$$\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 :