Moments for Joint Distributions

Study guide: Exam P-1

Moments for joint distributions generalize Expected Value to functions of multiple random variables. Covariance is computed from these moments as $\text{Cov}(X,Y) = E[XY] - E[X]E[Y]$ .

$E[X] = \sum_x \sum_y x\,p(x,y)$

$E[Y] = \sum_x \sum_y y\,p(x,y)$

$E[XY] = \sum_x \sum_y xy\,p(x,y)$

$E[g(X,Y)] = \sum_x \sum_y g(x,y)\,p(x,y)$

For Independent Random Variables, $E[XY] = E[X]E[Y]$ , so $\text{Cov}(X,Y) = 0$

Computing E[XY] Joint PMF: $p(0,0)=0.3$ , $p(0,1)=0.2$ , $p(1,0)=0.1$ , $p(1,1)=0.4$ .

Answer $E[XY] = (0)(0)(0.3) + (0)(1)(0.2) + (1)(0)(0.1) + (1)(1)(0.4) = 0.4$ $E[X] = 0.1 + 0.4 = 0.5, \quad E[Y] = 0.2 + 0.4 = 0.6$ $\text{Cov}(X,Y) = 0.4 - (0.5)(0.6) = 0.4 - 0.3 = 0.10$