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### 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*.

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## (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.

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## (b) Conditions for \(X\) and \(Y\) to be Independent:
Two variables \(X\) and \(Y\)

Suppose X and Y take values in {0,1} with joint mass function p(x, y). Write p(0,0) = a, p(0,1) = b, p(1,0) = c, p(1,1) = d. Find necessary and sufficient conditions on a, b, c, d for X, Y to be (a) uncorrelated (b) independent.

Explore the necessary and sufficient conditions for two random variables, X and Y, with joint mass function p(x, y), to be uncorrelated and independent. This problem covers key probability concepts such as joint PMFs, covariance, and conditions for independence. Suitable for undergraduate students.

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