Let X and Y be real-valued random variables, both distributed according to a distribution \, \mathbf{P}.\, (We make no assumption about their joint distribution). Let F denote the cdf of \, \mathbf{P}\,.

Question

Which of the following are true about the cdf F? (Choose all that apply.)

P(X \leq t) and P(Y \leq t) are random variables.

For all t \in \mathbb {R}, F(t) = P(X \leq t) and F(t) = P(Y \leq t).

F(t) = P(X \leq t) = P(Y \leq t) only if X and Y are independent.

\displaystyle \lim _{t \to \infty } F(t) = 1.

\displaystyle \lim _{t \to -\infty } F(t) = 0.

\displaystyle \int _{-\infty }^{\infty } F(t) dt =1
unanswered

Answer 1

The correct options are:

1. P(X <= t) and P(Y <= t) are random variables.
4. lim (t -> infinity) F(t) = 1.
5. lim (t -> -infinity) F(t) = 0.
6. ∫(-infinity to infinity) F(t) dt = 1.

Option 2 is incorrect because F(t) = P(X <= t) and F(t) = P(Y <= t) are both true, but they don't have to be equal.

Option 3 is incorrect because F(t) = P(X <= t) = P(Y <= t) does not necessarily imply that X and Y are independent.