# Probability Density Functions

Hi!

I am currently studying Probability Density Functions and I am having a hard time wrapping my head around something.

So, from what I have read, P(X=c), i.e. probability that the random variable X takes on any specific outcome, is equal to 0. Yet, the probability X takes on any outcome between a and b is not 0. Isn't the probability that X takes on any value between a and b equal to the probability X takes on each individual outcome between a and b added together? In other words, would not that be equal to summing several probabilities = 0?

I hope I made my doubt somewhat clear,

jbunniii
Homework Helper
Gold Member
So, from what I have read, P(X=c), i.e. probability that the random variable X takes on any specific outcome, is equal to 0. Yet, the probability X takes on any outcome between a and b is not 0. Isn't the probability that X takes on any value between a and b equal to the probability X takes on each individual outcome between a and b added together? In other words, would not that be equal to summing several probabilities = 0?
You are correct that if ##X## has a probability density function then ##P(X = c) = 0## for any specific outcome ##c##. Your confusion stems from the fact that if you consider an uncountably infinite set of outcomes, such as the interval ##[a,b]##, then you cannot simply add the probabilities of the individual points. This only works for a finite or countably infinite number of outcomes.

Forget about probability for a moment and consider the interval ##[0,1]##. This interval has length 1 even though each point in the interval has length zero. There's no contradiction here, it's just a fact of life: we can't add the lengths (or probabilities, or more generally, the measures) of an uncountably infinite number of objects.

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Got it! Thank you very much jbunniii!