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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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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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