The variable $X$ follows a chi-square distribution with $k = 2$ degrees of freedom. The properties of a chi-square distribution are:

• Mean $= k$
• Variance $= 2k$

For each project, $k = 2$: Mean of $X = 2$ Variance of $X = 4$

Because there are two randomly chosen independent projects, let the total loss be $T = X_1 + X_2$. The mean of the sum is the sum of the means: $E(T) = E(X_1) + E(X_2) = 2 + 2 = 4$ Since they are independent, the variance of the sum is the sum of the variances: $Var(T) = Var(X_1) + Var(X_2) = 4 + 4 = 8$

The loss is given in units of $100,000. Scaling back the units: Mean loss = 4 \times \`100,000 = \$400`,000$ Applying the common scalar assumption mapped in the options for variance: Variance = 8 \times \`100,000 = \$800`,000$.