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Machine Learning A Bayesian and Optimization Perspective (Chapter 02:…

- - - - establish common language and understood notation
  - - - a random variable is described in term of
        
        probabilities if its values are of a discrete nature
        
        probability density function (pdf) if its values lie anywhere within an interval of the real axis
      - a random variable is a variable whose variations are due to change/randomness
    - - Two commonly used definitions:
        
        Relative frequency definition:
        \( P(A) = \lim{x \to \infty} \frac{n_A}{n} \)
        In practice, we use:
        \( P(A) \approx \frac{n_A}{n} \)
        
        Axiomatic definition: is a nonnegative number assigned to this event: \( P(A) \geq 0 \)
      - Some properties:
        
        Probability of an event C, which is certain to occur, equals to one
        \( P(C) = 1\)
        
        Probability of an impossible event is equal to zero \( P(C) = 0 \)
        
        If two events, A and B, are mutually exclusive (they cannot occur simultaneously), then the probability of occurence of either A or B is given by: \( P(A \cup B) = P(A) + P(B) \)
    - - a discrete random variable, x, can take any value from a finite or infinite set \( \mathcal{X} \).
        
        the prob of the event, \( \mathrm{x} = x \in \mathcal{X} \), is denoted as: \( P(\mathrm{x} = x) \) or simply \( P(x) \)
        
        the function \( P(\cdot) \) is known as probability mass function (pmf)
        
        \( P(x) \geq 0 \)
        
        \( \sum_{x \in \mathcal{X}} P(x) = 1 \)
        
        The set \( \mathcal{X} \) is known as the sample or state space
      - Joint Probability of the events, A, B