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I have a set of identically distributed Weibull's (or pick any other dist.) and they are all perfectly correlated. Can I treat them as a single Weibull with the same parameters as as a single Weibull, in order to determine moments and probabilities?

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Usually, if all variables are perfectly correlated, only one variable is needed to explain the data because other variables are redundant, but if all variables are uncorrelated we need all the variables to explain the data.

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In some cases of uncorrelated but dependent random variables, one might not need all the variables. For example, if $X$ is a random variable with symmetric density (that is, $f_X(x) = f_X(-x)$) and with finite third moment, then $X$ and $Y = X^2$ are uncorrelated random variables, but we don't need $Y$ if we have $X$. –  Dilip Sarwate Oct 26 '12 at 4:28

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