PROBABILITIES Course

Conditional Distribution

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In French it's called Loi de probabilité conditionnelle, distribution conditionnelle or Loi conditionnelle. This is the probability of a variable taking a value, given that another variable took a value.


Discrete Conditional Distribution

The probability of $X=x$ given $Y=y$ is

@ \mathbb{P}(X = x\ |\ Y=y) = \frac{ \mathbb{P}(X = x \cap Y=y) }{ \mathbb{P}(Y=y) } @

We can add

\[ \mathbb{E}(X\ |\ Y=y)= \sum_{i=1}^{n} x_i * \mathbb{P}(X = x_i\ |\ Y = y) \]
\[ V(X|Y=y)= \sum_{i=1}^{n} (x_i-\mathbb{E}(X))^2 * \mathbb{P}(X = x_i\ |\ Y = y) \]

Continuous Conditional Distribution

The formula changes a bit for continuous variables

@ f_{X|Y}(x\ |\ y) = \frac{ f_{X|Y}(x,\ y) }{ f_{Y}(y) } @

We can add

\[ \mathbb{E}(X\ |\ Y=y)= \int x * f_{X|Y} (x|y) dx \]
\[ V(X\ |\ Y=y)= \int (x-\mathbb{E}(X))^2 * f_{X|Y} (x,y) dx \]