ements-is-true-regarding-the-probability-that-x-equals-an-exact-number-a-ie-px-a

Question

For a continuous random variable X with PDF f(x), which of the following statements is true regarding the probability that X equals an exact number a (i.e., P(X = a))?

Accepted Answer

P(X = a) = 0 for any single point a (assuming a continuous distribution without atoms)

Answer

P(X = a) = f(a) because the PDF evaluated at a gives the exact probability of X equaling that point

Answer

P(X = a) equals the integral of f over an infinitesimal interval around a, which is always a finite nonzero number identical to f(a) times an infinitesimal width

Answer

P(X = a) depends on whether a is rational or irrational; rational points can have positive probability while irrational points cannot in continuous models