Expectation, variance and moments

2 free worked examples

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1. Exam PExpectation, variance and momentsCore

   A random variable has $E[X] = 4$ and $E[X^{2}] = 25$. Find the variance.

   1. A$3.0000$
   2. B$5.0000$
   3. C$9.0000$
   4. D$16.0000$
   5. E$21.0000$

   Solution

   1. The computational formula is $\operatorname{Var}(X) = E[X^{2}] - (E[X])^{2}$.
   2. $(E[X])^{2} = 16$.
   3. $\operatorname{Var}(X) = 25 - 16$.
   4. $= 9$, so the standard deviation is 3. Never subtract $E[X]$ from $E[X^2]$ directly - the square matters.

   Trap. Reporting the standard deviation 3, or subtracting 4 from 25 to get 21.

2. Exam PExpectation, variance and momentsStretch

   A non-negative variable has survival function $S(x) = (1+x)^{-2}$ for $x>0$. Find $E[X]$.

   1. A$0.5000$
   2. B$1.0000$
   3. C$1.5000$
   4. D$2.0000$
   5. EIt does not exist

   Solution

   1. For a non-negative variable, $E[X] = \int_0^{\infty}S(x)\,dx$ - usually much faster than integrating $xf(x)$.
   2. $\int_0^{\infty}(1+x)^{-2}dx = \left[-(1+x)^{-1}\right]_0^{\infty}$.
   3. $= 0 - (-1) = 1$.
   4. The survival-function route avoids differentiating $S$ to get the density at all.

   Trap. Differentiating S to get f and then integrating xf(x) - correct but twice the work.