The given problem describes two random variables $X$ and $Y$ taking values in $\{0,1\}$, with the joint probability mass function $p(x,y)$. It defines the probabilities:

$p(0,0) = a, \quad p(0,1) = b, \quad p(1,0) = c, \quad p(1,1) = d$

(a) Uncorrelated Condition:

Two random variables $X$ and $Y$ are uncorrelated if their covariance is zero. The covariance is given by:

[ \text{Cov}(X,Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y] ]

Step 1: Compute $\mathbb{E}[X]$ and $\mathbb{E}[Y]$

$\mathbb{E}[X] = 0 \cdot (a + b) + 1 \cdot (c + d) = c + d$ $\mathbb{E}[Y] = 0 \cdot (a + c) + 1 \cdot (b + d) = b + d$

Step 2: Compute $\mathbb{E}[XY]$

$\mathbb{E}[XY] = 0 \cdot 0 \cdot a + 0 \cdot 1 \cdot b + 1 \cdot 0 \cdot c + 1 \cdot 1 \cdot d = d$

Step 3: Covariance formula

$\text{Cov}(X,Y) = d - (c+d)(b+d)$

Thus, for $X$ and $Y$ to be uncorrelated, the covariance must be zero:

$d - (c+d)(b+d) = 0$

(b) Independent Condition:

Two random variables $X$ and $Y$ are independent if the joint probability factorizes into the product of the marginal probabilities. This means:

$p(x,y) = p_X(x)p_Y(y)$

Step 1: Marginal probabilities

• Marginal probability of $X$: $p_X(0) = a + b, \quad p_X(1) = c + d$
• Marginal probability of $Y$: $p_Y(0) = a + c, \quad p_Y(1) = b + d$

Step 2: Factorization condition

For independence, we need: $p(0,0) = p_X(0)p_Y(0), \quad p(0,1) = p_X(0)p_Y(1), \quad p(1,0) = p_X(1)p_Y(0), \quad p(1,1) = p_X(1)p_Y(1)$

This gives the following system of equations: $a = (a+b)(a+c), \quad b = (a+b)(b+d), \quad c = (c+d)(a+c), \quad d = (c+d)(b+d)$

Summary:

• For $X$ and $Y$ to be uncorrelated, the condition is: $d = (c+d)(b+d)$
• For $X$ and $Y$ to be independent, the conditions are the system of four equations: $a = (a+b)(a+c), \quad b = (a+b)(b+d), \quad c = (c+d)(a+c), \quad d = (c+d)(b+d)$

Let me know if you want more details or have further questions!

Related Questions:

1. How do you compute the marginal probabilities from a joint probability table?
2. What is the difference between uncorrelated and independent random variables?
3. Can two random variables be uncorrelated but not independent?
4. How does the covariance between two random variables affect their correlation?
5. What are the properties of joint probability mass functions for discrete variables?

Tip: Independence implies uncorrelation, but the reverse is not always true. Be mindful when interpreting relationships between variables.