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Would it be safe to say that a random variable $X$ is identially zero when its first and second moments are both zero? If it is true, how would you prove this?

This step is needed when we prove that the correlation coefficient of $X$ and $Y$ is $\pm 1$ if and only if $X$ and $Y$ are linearly related.

BUNKER
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    If the mean and variance are 0 (my understanding of 1st and 2nd moments), then $X$ should be zero with probability 1. Depending on how much measure theory you've had (and whether this is a discrete or continuous RV) then you may know that occurring with probability 1 doesn't always mean it always occurs... just that the times when it doesn't occur have measure 0. – TravisJ Apr 29 '15 at 14:54
  • The second moment being zero suffices since $Y\geqslant0$ almost surely and $E(Y)=0$ imply that $Y=0$ almost surely. – Did Apr 29 '15 at 15:22

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We can say that $X = 0$ with probability $1$. Since $X$ is stochastic,we cannot say that $X$ is exactly $0$, but we can say the probability $X$ is different than $0$ is $0$

Determine $P(X ≠ 0) = P(|X|>\delta)\text{ } \forall \delta>0$

Now, fix arbitrary $\delta > 0$

Then, $P(|X|>\delta) < \frac {\Bbb E[X^2]}{\delta^2} = 0 \Rightarrow P(|X|>\delta) = 0 \text{ } \forall \delta>0 $

Where Markov's inequality was used to establish the first inequality.

Hence, $P(X ≠ 0) = 0 \Rightarrow X = 0$ with probability $1$

Jamie Lannister
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