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I have a very simple question. I heard in my class that a random variable is equal to zero if its variance is equal to zero. I understand that if the variance of a random variable is zero, then that random variable must be a constant. But if it is a continuous random variable, does that mean that the random variable is equal to zero (since it cannot take a particular value)?

Thank you.

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  • $\begingroup$ It is not clear what you are asking. If the variance is zero then the function has the same value ae. What do you mean by a continuous rv? Do you mean the function or its distribution? $\endgroup$ – copper.hat Feb 13 at 20:12
  • $\begingroup$ $X$ doesn't have to be zero if it has zero variance. If however $X$ has zero mean, then that would imply $X$ is degenerate at zero. (And if $X$ takes a constant value, how can it be continuous?) $\endgroup$ – StubbornAtom Feb 13 at 20:15
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    $\begingroup$ You heard wrong. A random variable is constant (not necessarily $0$) if its variance is $0$. If it is constant, it is not a continuous random variable. $\endgroup$ – Robert Israel Feb 13 at 20:15

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