if $X,Y,Z$ are random variables does $P(X\mid Y,Z)=P(X\mid Y)$ when $X$ and $Z$ are independent.
I thought it's true because $X$ is independent in $Z$, and therefore providing data about $Z$ does not affect the distribution of $X$, but I couldn't prove it. (I think I saw that in this case $X-Y-Z$ is a Markov chain so it should be true)
One more question: Assuming that {$Y = Y_1 \: \text{ if }\: Z=0; Y_2 \:\text{ if }\: Z=1$} where $Y_1$ and $Y_2$ are also random variables, can I write that: $P(X \mid Y, Z=0)= P(X\mid Y_1)$ and $P(X\mid Y, Z=1)= P(X \mid Y_2)$? it seems to me that if the statement above is true then I can't write it.
Sorry if my question is not so smart, sometimes these notations in probability makes me frustrated.