Characterise the set of symmetric probability mass functions Consider the stepwise cumulative distribution function
$$
\Delta(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\geq \mu_j\} \hspace{1cm} \forall x \in \mathbb{R}
$$
where


*

*$J<\infty$

*$\lambda\equiv (\lambda_1,...,\lambda_J)$

*$\mu\equiv (\mu_1,...,\mu_J)$

*$\mu_1<...<\mu_J$, $\mu_j\in \mathbb{R}$ $\forall j$

*$\lambda_j\in [0,1]^J$ and $\sum_{j=1}^J \lambda_j=1$

Let $P(\cdot; \lambda,\mu)$ denote the probability mass function associated with the CDF $\Delta(\cdot; \lambda,\mu)$.

Consider the set
$$
\Omega_J\equiv \{(\lambda, \mu): P(\cdot; \lambda,\mu) \text{ is symmetric}\}
$$

Question: is it possible to explicitly characterise the set $\Omega_J$ through conditions on $(\lambda, \mu)$ for any generic $J$?
For example, 
$$
\Omega_2=\Big\{(\lambda,\mu): \lambda_1=1\text{ and }\lambda_2=0\text{, or }\lambda_1=0\text{ and }\lambda_2=1\text{, or } \lambda_1=1/2\text{ and }\lambda_2=1/2\Big\}
$$
$$
\Omega_3=\Big\{(\lambda,\mu): \lambda_1=\lambda_3 \text{ and } \mu_2-\mu_1=\mu_3-\mu_2\text{, or } \lambda_j=0 \text{  for some $j\in \{1,2,3\}$} \Big\}
$$
I'm struggling to generalise these characterisations to any $J$. Any help?
 A: Here are the required conditions: Symmetric mass functions (over a finite support) can be characterised by symmetry of support values around a mean/median and equality of probability values reflected around that mean/median.  Thus, in order to have a symmetric mass function the vector $\boldsymbol{\mu}$ must satisfy the requirement:$^\dagger$
$$\mu_k - \mu_{k'} =  \mu_{J-k'+1} - \mu_{J-k+1}  \quad \quad \quad \text{for all } k,k' = 1,...,J,$$
and the vector $\boldsymbol{\lambda}$ must satisfy the requirement:
$$\lambda_k = \lambda_{J-k+1} \quad \quad \quad \text{for all } k = 1,...,J.$$
These two conditions exhaust all symmetric mass functions with a finite number of outcomes $J$.  The mean/median of the resulting mass function is:
$$\mathbb{E}(X) = \text{Median}(X) = \frac{\mu_k + \mu_{J-k+1}}{2} \quad \quad \quad \text{for all } k = 1,...,J.$$

$^\dagger$ This first condition can be written alternatively as:
$$(\exists \mu_* \in \mathbb{R}) (\forall k = 1,...,J): \quad \mu_k + \mu_{J-k'+1} = 2 \mu_*.$$
In this alternative statement of the condition we use the value $\mu_* = \mathbb{E}(X) = \text{Median}(X)$, so the condition explicitly invokes an existence condition on the mean/median.
