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Conditional Probability

3.1 Continuous Probability Distribution and Density Function

Continuous Probability Distributions

  • The probability distribution for a continuous random variable is a description all the possible values of the random variable, and its corresponding probability.
  • Unlike discrete random variables which takes on a set of individual values, a continuous random variable is one that can take any value in an interval of the real number line.

For example, X is a continuous random variable if it takes on any variable in the interval [-0.005,0.005]. In this case, X might be the standard error when measuring using a micrometer screw gauge.

  • A continuous random variable has no limit as to the accuracy with which it can be measured. Using the example above, we can have
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1.2 Introduction to Probabilities

Probabilities

  • Probability is a numerical measure of the chance of a particular event occurring.
  • It is defined that the sum of probabilities of all possible outcomes to be 1, and all individual probabilities is non-negative. To sum up, the probability of each outcome is a real number in the interval [0,\ 1].
  • The probability of an event is equal to the sum of the probabilities of the outcomes in that event.
  • Mathematically, we represent probability of an event as \operatorname{Pr}(X=x) or \operatorname{Pr}(x), which X is the criteria to be observed, and x is the (actual) possible outcome.
  • Usually, we assume that each outcome is equally likely to happen i.e. have the same probability, unless there are conditions/evidence that says otherwise. Thus, if there are n equally likely outcomes, probability of each event is \frac{1}{n}.
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