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Conditional probability distribution

In probability theory and statistics, given two jointly distributed random variables and , the conditional probability distribution of Y given X is the probability distribution of when is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value of as a parameter. When both and are categorical variables, a conditional probability table is typically used to represent the conditional probability. The conditional distribution contrasts with the marginal distribution of a random variable, which is its distribution without reference to the value of the other variable.

If the conditional distribution of given is a continuous distribution, then its probability density function is known as the conditional density function. The properties of a conditional distribution, such as the moments, are often referred to by corresponding names such as the conditional mean and conditional variance.

More generally, one can refer to the conditional distribution of a subset of a set of more than two variables; this conditional distribution is contingent on the values of all the remaining variables, and if more than one variable is included in the subset then this conditional distribution is the conditional joint distribution of the included variables.

Contents

Conditional discrete distributionsEdit

For discrete random variables, the conditional probability mass function of   given   can be written according to its definition as:

 

Due to the occurrence of   in a denominator, this is defined only for non-zero (hence strictly positive)  

The relation with the probability distribution of   given   is:

 

ExampleEdit

Consider the roll of a fair die and let   if the number is even (i.e. 2, 4, or 6) and   otherwise. Furthermore, let   if the number is prime (i.e. 2, 3, or 5) and   otherwise.

1 2 3 4 5 6
X 0 1 0 1 0 1
Y 0 1 1 0 1 0

Then the unconditional probability that   is 3/6 = 1/2 (since there are six possible rolls of the die, of which three are even), whereas the probability that   conditional on   is 1/3 (since there are three possible prime number rolls—2, 3, and 5—of which one is even).

Conditional continuous distributionsEdit

Similarly for continuous random variables, the conditional probability density function of   given the occurrence of the value   of   can be written as[1]:p. 99

 

where   gives the joint density of   and  , while   gives the marginal density for  . Also in this case it is necessary that  .

The relation with the probability distribution of   given   is given by:

 

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.

ExampleEdit

 
Bivariate normal joint density

The graph shows a bivariate normal joint density for random variables   and  . To see the distribution of   conditional on  , one can first visualize the line   in the   plane, and then visualize the plane containing that line and perpendicular to the   plane. The intersection of that plane with the joint normal density, once rescaled to give unit area under the intersection, is the relevant conditional density of  .

Relation to independenceEdit

Random variables  ,   are independent if and only if the conditional distribution of   given   is, for all possible realizations of  , equal to the unconditional distribution of  . For discrete random variables this means   for all possible   and   with  . For continuous random variables   and  , having a joint density function, it means   for all possible   and   with  .

PropertiesEdit

Seen as a function of   for given  ,   is a probability mass function and so the sum over all   (or integral if it is a conditional probability density) is 1. Seen as a function of   for given  , it is a likelihood function, so that the sum over all   need not be 1.

Measure-theoretic formulationEdit

Let   be a probability space,   a  -field in  , and   a real-valued random variable (measurable with respect to the Borel  -field   on  ). Given  , the Radon-Nikodym theorem implies that there is[2] a  -measurable integrable random variable   so that   for every  , and such a random variable is uniquely defined up to sets of probability zero. It can then be shown that there exists[3] a function   such that   is a probability measure on   for each   (i.e., it is regular) and   (almost surely) for every  . For any  , the function   is called a conditional probability distribution of   given  . In this case,

 

almost surely.

Relation to conditional expectationEdit

For any event  , define the indicator function:

 

which is a random variable. Note that the expectation of this random variable is equal to the probability of A itself:

 

Then the conditional probability given   is a function   such that   is the conditional expectation of the indicator function for  :

 

In other words,   is a  -measurable function satisfying

 

A conditional probability is regular if   is also a probability measure for all ω ∈ Ω. An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.

  • For the trivial sigma algebra   the conditional probability is a constant function,  
  • For  , as outlined above,  .

See alsoEdit

NotesEdit

  1. ^ Park,Kun Il (2018). Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer. ISBN 978-3-319-68074-3.
  2. ^ Billingsley (1995), p. 430
  3. ^ Billingsley (1995), p. 439

ReferencesEdit