Marginal Probability

The marginal probability is just the probability of one event in the presence of all outcome of the other random variables from which the event depends on.

Let’s say we know that the random variable $X$ depends on $Y$, which can assume the values $0,1$. we also know that $P(X,Y=0)=0.3$ and $P(X,Y=1)=0.1$, then the marginal probability $P(X)$ is: $P(X)={\sum }_i^{C}P(X,Y=i)$where $C=\{0,1\}$ is the set of all possible values of $Y$.

Note that with $P(X,Y)$ we denote the joint probability between $X$ and $Y$. $P(X,Y)$ can be computed as $P(X,Y)=P(X)P(Y|X)$ Just for completion, we can compute the joint probability between three random variables $X,Y,Z$ as: $P(X,Y,Z)=P(X)P(Y|X)P(Z|Y)$

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tags: probability-theory