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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?

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  • $\begingroup$ For $3$...suppose $F$ is the complement of $E$. $\endgroup$
    – lulu
    Nov 7, 2018 at 15:27
  • $\begingroup$ Why do you think $4$ is false? Have you ever tried to prove it? $\endgroup$
    – BGM
    Nov 7, 2018 at 15:32
  • $\begingroup$ 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? $\endgroup$
    – Lanie
    Nov 7, 2018 at 15:55

1 Answer 1

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  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 $\le2$.
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