Continuous Distributions - rFronteddu/general_wiki GitHub Wiki

These distributions deal with continuous values, this results in approaches that deal with curves instead of disconnected points.

  1. Normal Distribution: often observed in nature
  2. Student's T: Approximate Normal distribution with limited data. It accommodates extreme values better by having fatter tails.
  3. Chi-Squared: Asymmetric, non-neg, used in hypothesis testing to determine goodness of fit.
  4. Exponential: Describes events that rapidly change early on.
  5. Logistic: Used in forecasting analysis to determine a cut-off point for a successful outcome.

By having infinitely many consecutive outcomes, the sample space is infinite. We cannot thus record the frequency of each distinct value in a table. For this reason, we use a curve instead.

  • What we graph, is the Probability Density Function PDF = f(y) where y is an element of the sample space. The PDF gives the probability of each possible y in the sample space. Since it expresses a non-negative probability.
  • The graph created using f(y) is called the Probability Distribution Curve PDC since it shows the likelihood of each outcome.

Note that since the sample space is infinite, P(y) = 0 (this probability is defined as favored outcomes over sample space) in continuous distribution. That also means that P(x>X) = P(x>=X)

  • The Cumulative Distribution Function is the probability that a random variable Y is smaller or equal to a value
    • Y: CDF: F(Y) = P(Y⇐y)

Note that since no value can be bigger than infinite. F(∞) = 1. Conversely F(-∞) = 0

Note that discrete distributions have CDF too but they can be obtained by summing PDF values.

We can use the CDF to estimate the probability of some interval integrating the PDF in it. Similarly, we can extract the PDF by derivating the CDF.

For the expected value we use the same definition that we used for continuous distributions but then we integrate instead of summating.

E(X)=∫⁡x⋅f(x)dx from −∞ to ∞