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I am curios about the concept of Variance. I try to get the better understanding of the variance by checking extreme cases.

$Var(X) = E[(X^2)] - (E[X])^2$

question 1. What does it mean when Variance equals 0.

If variance of random variable X equals 0, it means that

$E[(X^2)] = (E[X])^2$

in case of discrete random variable, we can rewrite it as

$\sum_{x \in X}^{} p(x)x^2 = (\sum_{x \in X}^{} p(x)x)^2$

in case when sample set is just one element $x$ with probability 1, we get

$p(x)x^2 = p(x)^2x^2$, which is OK when $p(x)=1$.

In case when sample is two elements with equal probabilities (uniform distribution), variance doesn't equal 0.

So here I didn't answer my question when variance equals 0 (the only obvious case is when sample space consist of one element).

Do you know more cases when variance is 0?

question 2. What does it mean when variance equals 1.

Few times I saw the mentioning of unit variance (I assume is when variance equals 1)

the question is when $E[(X^2)] - (E[X])^2=1$.

I cannot come up with cases when variance is 1.

Can you explain what is so special about unit variance, and when it occurs?

Addendum:

Thanks to the comments, lets try reconsider the case when variance equals 0.

$E[(X-\mu)^2]=0$

$\sum_{x \in X}^{} p(x)(x-\mu)^2= 0$

it happens, when either $p(x)=0$, or $x=\mu$.

Therefore, in order to random variable has the variance 0, it should get the only value (which is of course will be equal to $\mu$), any other value occurs with probability 0.

Is this the right reasoning?

user16168
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    There is nothing special about unit variance. Why do you think there is? – Did Sep 16 '13 at 09:08
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    For the zero variance question, the formula $\operatorname{Var}(X)=E[(X-E[X])^2]$ should be more enlightening. – Anthony Carapetis Sep 16 '13 at 09:10
  • @Did, about unit variance I read when was considering Standard normal distribution, where pdf is ensured to have unity variance. – user16168 Sep 16 '13 at 09:12
  • Unit variance is no more special than zero expectation value. It's nice to work with, but carries no real significance. – Arthur Sep 16 '13 at 09:13
  • The variance is a measure of the spread of the values of a random variable. The larger the variance, the more values that $X$ attains that are further from the expectation of $X$. In particular, variance of $0$ means the random variable attains only one value. Very large variance means relative large number of values are far from the expectation. There is nothing special about variance of 1. – Ittay Weiss Sep 16 '13 at 09:14
  • I think it is more useful and intuitive to think about $Var(X)$ from the formula $Var(X) = E[(X - E[X])^2]$. The special case when $E[X] = 0$ may be more illuminating than when $Var(X) = 0$. – Tunococ Sep 16 '13 at 09:15

1 Answers1

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variance is $E(X-EX)^2$. So when variance is $0$ you can say that the random variable $X$ is identically equal to its expectation $EX$. That is $X$ is a degenerate random variable that takes the value $EX$ with probability $1$. There is no other extreme cases. Variance $=1$ is not an extreme case at all. Variance of a random variable is as large as you can think of. The larger the variance the greater will be the observation on that random variable dispersed (or spread) around the mean value.

QED
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