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Conditional Probability Distribution

Conditional Probability Distribution

To get a complete understanding of the conditional probability distribution, it is important to know what is conditional probability. The conditional probability is the probability of occurrence of an event (say event A), given that another event has already occurred (say event B). That is what is the probability of event A happening, given that event B has already occurred. Therefore, the conditional probability distribution of A given B is the probability distribution of A when A is already known to be a particular value. In some case, the conditional probability  distribution may be illustrated in form of an unspecified value, a of A as a parameter. Generally, the conditional probability distribution is represented by te following formula:

The conditional probability is read as, the probability of event A occurring, given that event B has already occurred. This is given by the proportion of probability of A and B, divided by the probability of event B.

Categories of Conditional Probability Distribution

There are two types of conditional probability distributions, conditional discrete and the conditional continuous distribution.

For the conditional discrete distributions, the conditional probability of Y given X=x could be expressed by the following function.

Due to the function of the denoinator (P(X=x), this distribution is only defined for non-zeros hence considered strictly positive. A good  example is a fair rolling of a die. Let X=1 be an even nuber (2, 4, and 6) and X=0 otherwise. Further, Let Y=1 be a prie number (2, 3, 5) and Y=0 otherwise. The results of a roll of a die for 6 ties are resented below.

Then, the unconditional probability that X=1 is 3/6 = 1/3. This is because among the 6 rolls of a fair die, three outcomes are even. Additionally, the conditional probability of X=1 given that Y=1 is 1/3. This is because among the three Prime number outcomes (Y=1 = 2, 3 and 5), there is only one outcome that is even = 2.

For the conditional continuous variables, the conditional probability of Y given that the value x of X is represented by the following formula.

The function  represents the joint density between event X and Y, while the denominator   represents the marginal density.

A perfect example of a continuous conditional probability distribution could be shown in the figure below.

The graph is a representation of a bivariate normal joint distribution of  variable X and Y. the question could be, what is the distribution of Y given that X=70. This could be achieved by first visualizing the line X=70 on the plane from the X-axis, and then fro the Y- axis, visualise that line perpendicular to the X,Y plane.