# Which statement is true or are they all false?

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?

• For $3$...suppose $F$ is the complement of $E$.
– lulu
Nov 7, 2018 at 15:27
• Why do you think $4$ is false? Have you ever tried to prove it?
– BGM
Nov 7, 2018 at 15:32
• I know that if X and Y are independent then P(X intersect Y) = P(X)*P(Y), but E[XY]-E[X]E[Y] is equal to covariance so I assume then if they are independent then the covariance would equal zero therefore E[XY] would equal E[X]E[Y] since they cancel each other out? Nov 7, 2018 at 15:55

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.
3. False as $$0=P(EF)\ne P(E)P(F)$$.
4. True. You only need moments of order $$\le2$$.