Conditional Probability Density Function

A Conditional Probability Density Function is a conditional probability function that is a probability density function.

• AKA: Conditional Density Model.
• Context:
  • It can be produced by a [[Conditional Probability Density Function Creation
• Example(s):
  • Gaussian Conditional Density Function.
  • Binomial Conditional Density Function.
  • …
• Counter-Example(s):
  • a Conditional Probability Mass Function.
  • a Joint Probability Density Function.
• See: Random Variable Vector, Bivariate Random Vector, Continuous Random Vector, Borel's Paradox, Continuous Random Variable, Marginal Density, Conditional Mean, Conditional Variance.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Conditional_probability_distribution#Continuous_distributions Retrieved:2016-1-8.
  • Similarly for continuous random variables, the conditional probability density function of Y given the occurrence of the value x of X can be written as : [math]\displaystyle{ f_Y(y \mid X=x) = \frac{f_{X, Y}(x, y)}{f_X(x)}, }[/math] where f_X,Y(x, y) gives the joint density of X and Y, while f_X(x) gives the marginal density for X. Also in this case it is necessary that [math]\displaystyle{ f_X(x)\gt 0 }[/math] .

    The relation with the probability distribution of X given Y is given by: : [math]\displaystyle{ f_Y(y \mid X=x)f_X(x) = f_{X,Y}(x, y) = f_X(x \mid Y=y)f_Y(y). }[/math] The concept of the conditional distribution of a continuous random variable is not as intuitive as it might seem: Borel's paradox shows that conditional probability density functions need not be invariant under coordinate transformations.

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