# Conditional Probability Density Function

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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):****Counter-Example(s):****See:**Random Variable Vector, Bivariate Random Vector, Continuous Random Vector, Borel's Paradox, Continuous Random Variable, Marginal Density, Conditional Mean, Conditional Variance.

## 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.

- Similarly for continuous random variables, the conditional probability density function of