5.4 Independence of Random Variables

Concept: $X$ and $Y$ are independent if the occurrence of one does not affect the probabilities of the other.

Mathematical Condition: $X$ and $Y$ are independent if and only if: $$ f(x,y) = f_X(x) \cdot f_Y(y) $$ (For all values of $x$ and $y$).

---

Comprehensive Example (2D Variables)

Scenario: A bag contains 2 Red balls and 3 Blue balls. Two balls are drawn without replacement. Let $X$ = Number of Red balls drawn. Let $Y$ = Number of Blue balls drawn.

Joint Table:

Step 2: Find Marginal Distributions

• Marginal $P(X)$: Sum the columns.
• $P(X=0) = 0.30, P(X=1) = 0.60, P(X=2) = 0.10$
• Marginal $P(Y)$: Sum the rows.
• $P(Y=0) = 0.10, P(Y=1) = 0.60, P(Y=2) = 0.30$

Step 3: Check Independence Are $X$ and $Y$ independent? Check if $P(X=1, Y=1) = P(X=1) \cdot P(Y=1)$.

• Joint: $P(1,1) = 0.60$
• Product of Marginals: $P(X=1) \times P(Y=1) = 0.60 \times 0.60 = 0.36$.
• Since $0.60 \neq 0.36$, the variables are Dependent.

Step 4: Conditional Probability Find the probability of getting 1 Red ball given that we know we got 1 Blue ball. $$ P(X=1 | Y=1) = \frac{P(X=1, Y=1)}{P(Y=1)} = \frac{0.60}{0.60} = 1.0 $$ (Interpretation: If you know you drew exactly 1 Blue ball, you are 100% certain the other ball was Red in this context).