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As far as I am concerned, probability distribution function is for discrete random variables while probability density function is for continuous random variables. To find the probability value of continuous random variable, we have to take the total area under the function which differ from discrete random variable, where we can take the probability value directly from the function.

But here's my confusion,

Let's say Z=X+Y. X is discrete and can take in value of -1 and 1. Y is Gaussian random variable. So,Z is Gaussian random variable with probability density function of p(Z)=p(Z|X=1)+p(Z|X=-1).

However when we try to find p(X=1|Z), it is equals to [p(Z|X=1)p(X=1)]/p(Z). My questions, how can we time p(Z|X=1) and p(X=1) since former is probability density function while the latter is probability distribution function? What's more, p(Z) is also a probability density function. In the end, what is p(X=1|Z), a probability density or distribution function?

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    @mike: i suppose you mean a *mixture* distribution. $Z$ as defined has no atoms. [perhaps my microscope is just not strong enough to see them?]2010-11-28
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    @learnwhatever: what you refer to as the 'probability distribution function' of $X$ is better referred to as its 'probability mass function' [pmf]. the terms 'distribution function' [df] and 'cumulative distribution function' [cdf] usually pertain to what raphael discusses in his answer - and make sense for both continuous and discrete random variables. 'probability density function' [pdf] does [as you wrote] apply to continuous random variables. with these naming conventions, there is no ambiguity about what is being referred to.2010-11-28
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    $p_Y(y)$, $p_Z(z)$ and $p_{Z|X=1}(z|X=1)$ are densities, so you have to integrate them to get a probability. $\Pr(X=1)$ and $\Pr(X=1|Z=z)$ are probabilities.2012-06-01

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