Which statement is true or are they all false?

Question

Which of the following statements are true?

1. If the covariance of two random variables is zero, the random variables are independent.
2. If X is a continuous random variable, the continuity correction is used to approximate probabilities pertaining to X with a discrete distribution.
3. If E and F are mutually exclusive events which occur with nonzero probability, E and F are independent.
4. If X and Y are independent random variables, then given that their moments exist and E[XY] exists, E[XY]=E[X]E[Y].

I know that 1 is false and I am pretty sure that 4 is false, but I am not sure about 2 and three. I do not know what they are talking about in number 3 when they say continuity correction. Is 3 false because even though they are mutually exclusive the event A would occur if event B did not occur?

Answer 1

1. False, e.g. $X=\cos2\pi t$, $Y=\sin2\pi t$, with $t$ uniform on $[0,1]$. In other words the coords of a circular distribution.
2. Not sure what 'the continuity correction' means. It's sometimes used to approximate a discrete distribution by a continuous one, to 'smooth out the gaps'.
3. False as $0=P(EF)\ne P(E)P(F)$.
4. True. You only need moments of order $\le2$.