1. **Problem Statement:** Given the joint pmf of discrete random variables $X$ and $Y$: $$\begin{array}{c|ccc|c} X \backslash Y & 0 & 1 & 3 & f_1(x) \\ \hline 0 & \frac{1}{8} & \frac{2}{8} & \frac{1}{8} & \frac{4}{8} \\ 1 & \frac{2}{8} & \frac{1}{8} & \frac{1}{8} & \frac{4}{8} \\ \hline f_2(y) & \frac{3}{8} & \frac{3}{8} & \frac{2}{8} & 1 \end{array}$$ Find: (i) $E(Y|X=x)$ (ii) $\mathrm{Var}(Y|X=x)$ (iii) $\mathrm{Cov}(X,Y)$ (iv) Correlation coefficient $\rho_{XY}$ 2. **Formulas and Important Rules:** - Conditional expectation: $E(Y|X=x) = \sum_y y P(Y=y|X=x)$ - Conditional variance: $\mathrm{Var}(Y|X=x) = E(Y^2|X=x) - [E(Y|X=x)]^2$ - Covariance: $\mathrm{Cov}(X,Y) = E(XY) - E(X)E(Y)$ - Correlation coefficient: $\rho_{XY} = \frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}}$ 3. **Calculate $E(Y|X=x)$:** - For $X=0$: $$P(Y=y|X=0) = \frac{P(X=0,Y=y)}{P(X=0)}$$ Values: $P(Y=0|0) = \frac{1/8}{4/8} = \frac{1}{4}$, $P(Y=1|0) = \frac{2/8}{4/8} = \frac{1}{2}$, $P(Y=3|0) = \frac{1/8}{4/8} = \frac{1}{4}$ $$E(Y|0) = 0 \times \frac{1}{4} + 1 \times \frac{1}{2} + 3 \times \frac{1}{4} = 0 + \frac{1}{2} + \frac{3}{4} = \frac{5}{4} = 1.25$$ - For $X=1$: $$P(Y=0|1) = \frac{2/8}{4/8} = \frac{1}{2}, P(Y=1|1) = \frac{1/8}{4/8} = \frac{1}{4}, P(Y=3|1) = \frac{1/8}{4/8} = \frac{1}{4}$$ $$E(Y|1) = 0 \times \frac{1}{2} + 1 \times \frac{1}{4} + 3 \times \frac{1}{4} = 0 + \frac{1}{4} + \frac{3}{4} = 1$$ 4. **Calculate $\mathrm{Var}(Y|X=x)$:** - Compute $E(Y^2|X=x)$ first. - For $X=0$: $$E(Y^2|0) = 0^2 \times \frac{1}{4} + 1^2 \times \frac{1}{2} + 3^2 \times \frac{1}{4} = 0 + \frac{1}{2} + \frac{9}{4} = \frac{11}{4} = 2.75$$ $$\mathrm{Var}(Y|0) = 2.75 - (1.25)^2 = 2.75 - 1.5625 = 1.1875$$ - For $X=1$: $$E(Y^2|1) = 0^2 \times \frac{1}{2} + 1^2 \times \frac{1}{4} + 3^2 \times \frac{1}{4} = 0 + \frac{1}{4} + \frac{9}{4} = \frac{10}{4} = 2.5$$ $$\mathrm{Var}(Y|1) = 2.5 - (1)^2 = 2.5 - 1 = 1.5$$ 5. **Calculate $\mathrm{Cov}(X,Y)$:** - Find $E(X)$: $$E(X) = 0 \times \frac{4}{8} + 1 \times \frac{4}{8} = \frac{4}{8} = 0.5$$ - Find $E(Y)$: $$E(Y) = 0 \times \frac{3}{8} + 1 \times \frac{3}{8} + 3 \times \frac{2}{8} = 0 + \frac{3}{8} + \frac{6}{8} = \frac{9}{8} = 1.125$$ - Find $E(XY)$: $$E(XY) = \sum_x \sum_y xy P(X=x,Y=y)$$ $$= 0 \times (0 \times \frac{1}{8} + 1 \times \frac{2}{8} + 3 \times \frac{1}{8}) + 1 \times (0 \times \frac{2}{8} + 1 \times \frac{1}{8} + 3 \times \frac{1}{8})$$ $$= 0 + 1 \times (0 + \frac{1}{8} + \frac{3}{8}) = \frac{4}{8} = 0.5$$ - Therefore, $$\mathrm{Cov}(X,Y) = 0.5 - (0.5)(1.125) = 0.5 - 0.5625 = -0.0625$$ 6. **Calculate $\rho_{XY}$:** - Find $\mathrm{Var}(X)$: $$E(X^2) = 0^2 \times \frac{4}{8} + 1^2 \times \frac{4}{8} = 0 + \frac{4}{8} = 0.5$$ $$\mathrm{Var}(X) = E(X^2) - [E(X)]^2 = 0.5 - (0.5)^2 = 0.5 - 0.25 = 0.25$$ - Find $\mathrm{Var}(Y)$: $$E(Y^2) = 0^2 \times \frac{3}{8} + 1^2 \times \frac{3}{8} + 3^2 \times \frac{2}{8} = 0 + \frac{3}{8} + \frac{18}{8} = \frac{21}{8} = 2.625$$ $$\mathrm{Var}(Y) = 2.625 - (1.125)^2 = 2.625 - 1.265625 = 1.359375$$ - Finally, $$\rho_{XY} = \frac{-0.0625}{\sqrt{0.25 \times 1.359375}} = \frac{-0.0625}{\sqrt{0.33984375}} = \frac{-0.0625}{0.583} \approx -0.1072$$ **Final answers:** - (i) $E(Y|X=0) = 1.25$, $E(Y|X=1) = 1$ - (ii) $\mathrm{Var}(Y|X=0) = 1.1875$, $\mathrm{Var}(Y|X=1) = 1.5$ - (iii) $\mathrm{Cov}(X,Y) = -0.0625$ - (iv) $\rho_{XY} \approx -0.1072$