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I need to find conditional probability to count mutual information. Random variable X has uniform distribution on set {0, 1, 2}. Variable Z has Bernoulli distribution with parameter p=1/10. Y = X # Z, where # is "sum mod 3". I don't undestand the way how to count conditional probabilities.

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  • $\begingroup$ What do you mean by "alternative distribution"? $\endgroup$
    – Math1000
    Jan 26 '15 at 19:41
  • $\begingroup$ @Math1000 Sorry, Bernoulli distribution. $\endgroup$ Jan 26 '15 at 19:50
  • $\begingroup$ Are $X$ and $Z$ independent? $\endgroup$
    – Math1000
    Jan 26 '15 at 20:01
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We have

$$Y = \begin{cases} 0,& (X,Z)=(0,0) \text{ or } (X,Z)=(2,1)\\ 1,& (X,Z)=(0,1) \text{ or } (X,Z)=(1,0)\\ 2,& (X,Z)=(1,1) \text{ or } (X,Z)=(2,0). \end{cases} $$ So you can compute the distribution of $Y$ using the joint distribution of $Z$. For example if $X$ and $Z$ are independent then $$\mathbb P(Y=0) = \mathbb P(X=0)\mathbb P(Z=0) +\mathbb P(X=2)\mathbb P(Z=1). $$

If you wanted to compute e.g. $\mathbb P(Y=0|Z=0)$, then using the definition of conditional probability $$\mathbb P(Y=0|Z=0) = \frac{\mathbb P(Y=0,Z=0)}{\mathbb P(Z=0)}=\frac{\mathbb P(X=0,Z=0)}{\mathbb P(Z=0)}. $$

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  • $\begingroup$ Thank you. And what is P(Y=0, Z=0)? What does that comma mean? $\endgroup$ Jan 26 '15 at 21:31
  • $\begingroup$ The probability that $Y=0$ and $Z=0$. It's shorthand for $\mathbb P(\{Y=0\}\cap\{Z=0\})$. $\endgroup$
    – Math1000
    Jan 26 '15 at 21:41

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