Joint PMF of 2 discrete random variables

Question

Problem Let $\alpha_1,...,\alpha_n,\beta_1,...,\beta_n \in \mathbb{R}$, $\sum \alpha_i^2=\alpha$,$\sum \beta_i^2=\beta$, and $\sum \alpha_i\beta_i=\gamma$. If $\Delta=\alpha\beta-\gamma^2>0$, find the density of the joint distribution of $\alpha_1 X_1+...+\alpha_{n}X_n$ and $\beta_1 X_1+...+\beta_{n}X_n$ where $X_i$ are independent identically distributed Bernoulli $\pm 1$ random variables which take value $1$ with probability $0.5$ and value $-1$ with probability $0.5$.

Background The case where the $X_i$ are $N(0,1)$ appears in Lemma 4 of the paper On The Average Number of Real Roots of a Random Algebraic Equation.

Approach Let $\chi:=\alpha_1 X_1+...+\alpha_{n}X_n$ and $\mathcal{Y}:=\beta_1 X_1+...+\beta_{n}X_n$.

First, we must find $\mathbb{P}[\chi=x]$ and $\mathbb{P}[\mathcal{Y}=y]$. We could use convolutions inductively?

Then we need $\mathbb{P}[\chi = x \ and \ \mathcal{Y}=y]=\mathbb{P}[\chi=x | \mathcal{Y}=y]*\mathbb{P}[\mathcal{Y}=y]$

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Any help with this problem is immensely appreciated.

Answer 1

Let $a=(\alpha_1,\alpha_2,\ldots,\alpha_n)^{T}$ and $b = (\beta_1, \beta_2, \ldots, \beta_n)^T$.

Sample space of $\chi$ is $\{v^{T}a\text{ : } v \text{ is a binary n-vector with 0's replaced with -1's}\}$.

Sample space of $\mathcal{Y}$ is $\{v^{T}b\text{ : } v \text{ is a binary n-vector with 0's replaced with -1's}\}$.

Sample space of $(\chi,\mathcal{Y})$ is $\{(v^{T}a,v^{T}b)\text{ : } v \text{ is a binary n-vector with 0's replaced with -1's}\}$.

$$P(\chi=v_{0}^Ta, \mathcal{Y}=v_{0}^Tb) = P(X_1=v_{01}, X_2=v_{02}, \ldots, X_n=v_{0n}) = \prod_{i=1}^{n}P(X_i=v_{0i}) = \left(\frac{1}{2}\right)^n$$

For a fixed $x$ and $y$, let $V = \{v_{k} \text{ : } v_k^Ta = x \text{ and } v_k^Tb = y\}$, then,

$$P(\chi=x, \mathcal{Y}=y) = \sum_{v_{k}\in V} P(\chi=v_k^Ta,\mathcal{Y}=v_k^tb) = \sum_{v_k\in V}\left(\frac{1}{2}\right)^n = \frac{|V|}{2^n}$$

I am not sure if a better formulation of joint pmf is possible based on the condition OP wrote in the question.