# Latest content was relocated to https://bintanvictor.wordpress.com. This old blog will be shutdown soon.

## Thursday, June 2, 2011

### Prob(X=x) == 0 for any real-valued x

Mind the notation -- the big X __denotes__ a random variable such as the "angle" between 2 hands on a clock. The small x denotes a particular value of X, such as 90 degrees.

Discrete case -- we say P(X=head) = P(X=tail) = 0.5, or P(X=1 dot) = P(X=6 dots) = 1/6 on a dice. Always, all histogram lengths add up to 1.0.

When X is a continuous random var, then P(X=x) = 0 for any x, as seen in standard literature. This sounds bizarre, counter-intuitive and nonsensical -- the 2 hands did form a 90 degree, so 90 degree clearly isn't impossible.

A: Well, P(X=...) is undefined for a continuous RV. Range probability is well defined. If a range prob P(X>360) = 0, it does mean "impossible". When you see P(X=...) = 0 it means nothing -- no well-defined meaning.

That's a good enough answer for most of us. For the adventurous, If you really want a definition of P(X=..), then I'd say P(X=90) is defined as limit of P( 90< X <90+a ) as a approaches 0. Based on this definition, P(X=..anything) = 0