Find the maximum and minimum value of variance. [duplicate]

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Let $X$ be an arbitrary random variable takes values in $\{0,1,2,...,10\}$. Then the minimum and maximum values of variance of random variable $X$ are

1. $0$ and $30$
2. $1$ and $30$
3. $0$ and $25$
4. $1$ and $25$

But I don't know which one is true and how to approach? Help me out please.

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1 Answer

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For the minimal value of variance, think about the very word variance. Variance measures how much a random variable varies in its values. So, intuitively, a random variable with low variance does not vary a lot, i.e. it sticks to the same value most of the time. What kind of function (not random variable, simply function) would you say varies the least?

For the maximum, to make a variable with large variance, you need a variable that swings very much between two very different values, so you want a lot of values to be either $0$ or $10$...

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