What is the Paradox of Probability Density Functions?

[Peter G.]

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,

Thank you in advance!

 

[jbunniii]

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.

 

[Peter G.]

Got it! Thank you very much jbunniii!