Moments for Joint, Conditional, and Marginal Discrete Distributions

If large wind claims tend to coincide with large water claims, $E[XY]$ exceeds $E[X]\cdot E[Y]$, and the reserve must reflect the dependence. Computing moments from joint, conditional, and marginal distributions is how you capture these interactions.

For a function $g(X,Y)$:

$$E[g(X,Y)]=\sum _x\sum _{y}g(x,y)\cdot p_{X,Y}(x,y)$$

HIGH-FREQUENCY: The most common applications:

• $g=X$: gives $E[X]$ (equivalent to using the marginal)
• $g=XY$: gives $E[XY]$, needed for covariance
• $g=X^2$: gives $E[X^2]$, needed for variance

The conditional expected value of $Y$ given $X=x$:

$$E[Y|X=x]=\sum _{y}y\cdot p_{Y|X}(y|x)$$

This is a function of $x$. Different conditioning values give different answers. When viewed as a function of the random variable $X$, write $E[Y|X]$, this is itself a random variable.

The conditional variance:

Common mistakes

• Computing $E[XY]$ as $E[X]\cdot E[Y]$. This works only under independence. In Example 2: $E[X]E[Y]=(5/3)(47/120)=235/360$, but $E[XY]=97/120=291/360$. Trap: 235/360.
• Using the marginal variance formula for conditional variance. $\text{Var}(Y|X=x)$ uses the conditional PMF, not the marginal. These differ when $X$ and $Y$ are dependent.
• Forgetting the between-group term in Eve's law. Total variance is NOT just the average conditional variance. Omitting $\text{Var}(E[Y|X])$ gives 467/2400 = 0.1946 instead of the correct 0.2383. Trap: 0.1946.

Bottom line

• $E[g(X,Y)]={\sum }_x{\sum }_{y}g(x,y)\cdot p_{X,Y}(x,y)$: double sum over the joint PMF
• Iterated expectation: $E[Y]=E[E[Y|X]]={\sum }_{x}E[Y|X=x]\cdot p_X(x)$
• $E[Y|X]$ is a random variable (a function of X); only $E[E[Y|X]]$ collapses to the number $E[Y]$
• Eve's law: $\text{Var}(Y)=E[\text{Var}(Y|X)]+\text{Var}(E[Y|X])$, total = within-group + between-group variance

Exam shortcut

When you see a two-stage model ("Given $X$, $Y$ has distribution..."), immediately write $E[Y|X=x]$ and $\text{Var}(Y|X=x)$ for each $x$. These are sufficient for iterated expectations and Eve's law. Eve's Law. "EVVE": Expected Variance + Variance of Expectation = Total variance. "Independence kills the cross-term." If independent, $E[XY]=E[X]E[Y]$ and $\text{Cov}=0$. If dependent, compute $E[XY]$ from the joint.

The full lesson (about 3,453 words, 23 min read) adds 4 worked examples, all 7 common mistakes, a self-check, free in the app.

Learning objectives

• 3c

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