Conditional probability distribution

In probability theory and statistics, the conditional probability distribution is a probability distribution that describes the probability of an outcome given the occurrence of a particular event. Given two jointly distributed random variables and , the conditional probability distribution of given 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.[1] 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.

Conditional discrete distributions

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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 the denominator, this is defined only for non-zero (hence strictly positive)  

The relation with the probability distribution of   given   is:

 

Example

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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.

D 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 dice, 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 distributions

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Similarly for continuous random variables, the conditional probability density function of   given the occurrence of the value   of   can be written as[2]: 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.

Example

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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 independence

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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  .

Properties

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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 (or integral) over all   need not be 1.

Additionally, a marginal of a joint distribution can be expressed as the expectation of the corresponding conditional distribution. For instance,  .

Measure-theoretic formulation

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Let   be a probability space,   a  -field in  . Given  , the Radon-Nikodym theorem implies that there is[3] a  -measurable random variable  , called the conditional probability, such that for every  , and such a random variable is uniquely defined up to sets of probability zero. A conditional probability is called regular if   is a probability measure on   for all   a.e.

Special cases:

  • For the trivial sigma algebra  , the conditional probability is the constant function  
  • If  , then  , the indicator function (defined below).

Let   be a  -valued random variable. For each  , define  For any  , the function   is called the conditional probability distribution of   given  . If it is a probability measure on  , then it is called regular.

For a real-valued random variable (with respect to the Borel  -field   on  ), every conditional probability distribution is regular.[4] In this case,  almost surely.

Relation to conditional expectation

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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:

 

Given a  -field  , the conditional probability   is a version of the conditional expectation of the indicator function for  :

 

An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.

Interpretation of conditioning on a Sigma Field

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Consider the probability space   and a sub-sigma field  . The sub-sigma field   can be loosely interpreted as containing a subset of the information in  . For example, we might think of   as the probability of the event   given the information in  .

Also recall that an event   is independent of a sub-sigma field   if   for all  . It is incorrect to conclude in general that the information in   does not tell us anything about the probability of event   occurring. This can be shown with a counter-example:

Consider a probability space on the unit interval,  . Let   be the sigma-field of all countable sets and sets whose complement is countable. So each set in   has measure   or   and so is independent of each event in  . However, notice that   also contains all the singleton events in   (those sets which contain only a single  ). So knowing which of the events in   occurred is equivalent to knowing exactly which   occurred! So in one sense,   contains no information about   (it is independent of it), and in another sense it contains all the information in  .[5]

See also

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References

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Citations

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  1. ^ Ross, Sheldon M. (1993). Introduction to Probability Models (Fifth ed.). San Diego: Academic Press. pp. 88–91. ISBN 0-12-598455-3.
  2. ^ Park, Kun Il (2018). Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer. ISBN 978-3-319-68074-3.
  3. ^ Billingsley (1995), p. 430
  4. ^ Billingsley (1995), p. 439
  5. ^ Billingsley, Patrick (2012-02-28). Probability and Measure. Hoboken, New Jersey: Wiley. ISBN 978-1-118-12237-2.

Sources

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