I am writing a piece of work and have a situation where I have a 'double' conditional.

e.g. The event of Y = y conditional on X = x; the event X = x is also conditional on parameter z.

What's the convention to write this concisely?

$$P(Y = y \ | \ X|z = x)$$

$$P(Y = y \ | \ X = x ; z)$$

  • $\begingroup$ How exactly is the random variable $X$ conditional on the constant $z$? Of what is $z$ a parameter? $\endgroup$ – Graham Kemp Mar 31 '15 at 12:25
  • $\begingroup$ A suitable example would be X is a binomial random variable and z is the number of trials. $\endgroup$ – Brian Mar 31 '15 at 12:38

Suppose random variable $X$ has a probability distribution that is dependent on some parameter $z$.

Then we might write $\mathsf P(X=x; z) \mathop{:=} f_X(x;z)$ as long as we've established what the parameter $z$ means.

Then a conditional probability of a second random variable, $Y$ with respect to $X$ could be written $\mathsf P(Y=y\mid X=x; z)\mathop{:=}f_{Y\mid X}(y \mid x;z)$.

Note: $Y$ itself may, or may not, have a distribution dependent on the parameter $z$.


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