Monday, November 3, 2014

X vs x in probability formula, some basics to clarify

Stat risk question 1.1 -- For X ~ N(0,1), given a<X<b, there's some formula for the conditional CDF and PDF for X.

There's a bit of subtlety in the formula. When looking at the (sometimes complicated) formulas, it is useful to bear in mind and see through the mess that

- X is THE actual random variable (rvar), like the N@T
- The a and b are constant parameters, like 1.397 and 200
- x is the x-axis variable or the __scanner_variable__. For every value of x, we want to know exactly how to evaluate Pr(X < x).

X is a special species. You never differentiate against it. Any time you take sum, log, sqrt ... on it, you always, always get another rvar, with its own distribution.

Further, In a conditional density function f_X|y (x) = log(0.5+3+y ) + exp(x+y)/2π x, we had better look beyond the appearance of "function of two input variables". The two input variables have different roles
- x is the __scanner_variable__ representing the random variable X
- y is like the constant parameters a and b. At each value of y like y=3.12, random variable X has a distinct distribution.
- Note a constant parameter in a random variable distribution "descriptor" could be another random variable. From the notation f_X|y (x), we can't tell if y is (the scanner of) another rvar, though we often assume so.
- Notice the capital X in "X|y".

It's often helpful to go back to the discrete case, as explained very clearly in [[applied stat and prob for engineers]]

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