What is a conditional probability distribution

A conditional probability distribution is a probability distribution of Y that assumes that there are random variables X and Y and that X is a particular value. In other words, it is a probability distribution where event Y occurs under the assumption that event X has occurred. For example, if you roll the dice twice and get a 6 on the first roll, the conditional probability is the probability that the sum of the rolls will be 9.

Conditional probability distribution is represented as P(Y\mid X) or P_X(Y).

Conditional discrete probability distribution

For a discrete random variable, the conditional probability distribution is defined by the following equation:

{\displaystyle P(Y\mid X)={\frac {P(X \cap Y)}{P(X)}}}

P(X,Y) is the joint probability distribution of X and Y and P(X) is the marginal probability distribution.

As an example, suppose a couple has two children and at least one of them is known to be a girl. In this case, the probability that both children are girls is determined.

The probability that at least one of them is a girl is \frac{3}{4}.

P(X) = \frac{3}{4}

The probability that they are both girls is \frac{1}{4}.

P(X \cap Y) = \frac{1}{4}

Thus, the probability that they are both girls is \frac{1}{3}.

P(Y\mid X)={\frac {P(X \cap Y)}{P(X)}} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3}

Conditional continuous probability distribution

For a continuous random variable, the conditional probability distribution is defined by the following equation:

{\displaystyle f(y\mid x)={\frac {f(x,y)}{f(x)}}}, \quad f(x)>0

f(x,y) is the joint probability distribution of X and Y, and f(x) is the marginal probability distribution.