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I have a very basic question on when a discrete distribution might be called a symmetric distribution. Let say I have a r.v. $X$ that can take two possible values $(x1, x2)$ with $x1 \neq x2$ and corresponding probabilities $(0.4, 0.6)$. Then can I say that $X$ is symmetric?

Thanks,

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No, it only would be symmetric if the corresponding probabilities were (0.5,0.5).

Also, with binary or categorical distributions, the concept of symmetry does not have much meaning.

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  • $\begingroup$ Hi Hammer, can you please elaborate why it is not meaningful for discrete random variable? An example is highly appreciated. Thanks, $\endgroup$ – Saurav May 10 '11 at 18:20
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    $\begingroup$ Symmetry is not meaningful for a categorical random variable, because there is no natural ordering. Even if there is ordering, usually we don't know anything about spacing, so it is unclear which category should correspond to which "symmetric" one. But there are discrete distributions where symmetry makes sense, eg binomial or poisson distributions. $\endgroup$ – Aniko May 10 '11 at 18:45
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To elaborate on @benhammers answer:

Usually, a distribution is said to be symmetric if it is symmetric about its mean. In other words, the density (or probability function) should be symmetric around the mean $\mu=E(X)$. A discrete distribution is symmetric if $P(X=\mu-k)=P(X=\mu+k)$ for all values of $k$.

In your example, where $X$ takes the values $x_1$ and $x_2$ with probabilities $p_1$ and $p_2$, respectively, $\mu=E(X)=p_1x_1+p_2x_2$. For a two-point distribution, symmetry is only obtained when $p_1=p_2=1/2$.

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All right! In my opinion, it's meaningful to look at the symmetry of discrete random variables. As to how to characterize the symmetry of a discrete random variable, I think that for an multinomial random variable, firstly the values it takes must be symmetric about one point and then the corresponding probabilities are equal. Like for instance, random variable $X=\{1,2,\cdots,c\}$, and $P\{X=i\}=P\{X=c+1-i\}$ for all $i=1,2,\cdots,c$.

For other kinds of distributions, such as binomial distribution $B(n,p)$, I suppose that it's a symmetric distribution only if $p=\frac{1}{2}$. Actually, it's just my idea, maybe it's wrong. Hope others tell me their ideas.

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