Q120: A fair coin is tossed $n$ times. If $X$ and... - Numerical Analysis & Statistical Methods - Computer Application- ICAR JRF/SRF 2024 | Agrigyan

• Correlation Between Complementary Counts in Coin Tosses:
• In $n$ tosses of a coin, every toss results in either a head or a tail:

$$X + Y = n \implies Y = n - X$$

• The random variable $Y$ is an exact deterministic linear function of $X$ with a strictly negative slope:

$$Y = -1 \cdot X + n$$

• Correlation Coefficient Calculation:

$$\text{Cov}(X, Y) = \text{Cov}(X, n - X) = -\text{Var}(X)$$ $$\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} = \frac{-\sigma_X^2}{\sigma_X \cdot \sigma_X} = -1$$

• Because $X$ and $Y$ have a perfect inverse linear relationship, the correlation coefficient is identically -1.
• Conclusion: The correlation is -1 (Option 4).

Correct option: (4) A