Let's break down the given problem.

Problem Summary:

• $X$ and $Y$ are binary random variables that can take values from $\{0, 1\}$.
• The joint probability mass function (PMF) is denoted by $p(x, y)$.
• Given joint probabilities: $p(0, 0) = a, \quad p(0, 1) = b, \quad p(1, 0) = c, \quad p(1, 1) = d.$ We are tasked with determining the necessary and sufficient conditions on the parameters $a, b, c,$ and $d$ such that:

1. (a) $X$ and $Y$ are uncorrelated.
2. (b) $X$ and $Y$ are independent.

(a) Conditions for $X$ and $Y$ to be Uncorrelated:

Two random variables $X$ and $Y$ are uncorrelated if the covariance between them is zero: [ \operatorname{Cov}(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y] = 0. ]

1. Expectation Calculations:

   • $\mathbb{E}[X] = p(1, 0) + p(1, 1) = c + d$.
   • $\mathbb{E}[Y] = p(0, 1) + p(1, 1) = b + d$.
   • $\mathbb{E}[XY] = (1 \cdot 1) p(1, 1) = d$ (since only $(X, Y) = (1, 1)$ contributes).

2. Covariance Condition: Substituting these into the covariance formula: $\operatorname{Cov}(X, Y) = d - (c + d)(b + d).$ For $X$ and $Y$ to be uncorrelated, the covariance must be zero: $d - (c + d)(b + d) = 0 \quad \implies \quad d = (c + d)(b + d).$ This equation gives the condition for $X$ and $Y$ to be uncorrelated.

(b) Conditions for $X$ and $Y$ to be Independent:

Two variables $X$ and $Y$