• Definition of probability
  • Experiment
  • Sample space:
  • Event
  • Probability of an event:
    • a number assigned to an event $p(A)$
    • Axiom 1: $p(A) \geq 0$
    • Axiom 2: $p(S) = 1$
    • Axiom 3: For every sequence of disjoint events
      • $p(\bigcup_iA_i) = \sum_ip(A_i)$
• Variances
• Gaussian Parameter Estimation
  • We are given a data set of $N$ observations $x = x_1, x_2,...,x_n$ of the scalar variable $x$
  • Maximum Likelihood
    • Determines valued for the unknown parameters in the Gaussian by maximizing the likelihood function
    • Gaussian distribution mean is the same as the sample mean
    • $P(D)= P(x_1,x_2,...,x_n) = P(x_1) * p(x_2)* ...* p(x_n) = \Pi^N_{i=1}P(x_i)$
  • Likelihood function
    • $p(x|\mu,\sigma^2) = \Pi^N_{n=1}N(x_n|\mu,\sigma^2)$
    • To convenient the calculation, take $\log$ of both sides
  • When taking derivative, you know that the point where the derivative is 0 will be the maximum because the summation function is negative, resulting in an upside down parabola