Random Variables Questions and Answers

Answer: B

The expected value is 209.

Work out \((2x + 1)^{2}\) for each value and weight it by its probability: x = −3 gives 25, x = 6 gives 169, and x = 9 gives 361.

\[ \begin{aligned} E &= 25\cdot\tfrac{1}{6} + 169\cdot\tfrac{1}{2} + 361\cdot\tfrac{1}{3} \\ &= \frac{25 + 507 + 722}{6} \\ &= \frac{1254}{6} = 209 \end{aligned} \]

A common slip is to square \(2E(x) + 1\) instead: \(E(x) = 6\), giving \(13^{2} = 169\), which is not an option. 189, 219 and 229 come from arithmetic errors in the weighted sum.

Answer: D

The quartiles are 30 and 70.

Bowley's coefficient of skewness, based on quartiles, is:

\[ S_{k} = \frac{Q_{3} + Q_{1} - 2M}{Q_{3} - Q_{1}} \]

\[ \begin{aligned} 0.6 &= \frac{100 - 2(38)}{Q_{3} - Q_{1}} \\ &= \frac{24}{Q_{3} - Q_{1}} \\ Q_{3} - Q_{1} &= 40 \end{aligned} \]

With \(Q_{3} + Q_{1} = 100\), adding and subtracting gives \(Q_{3} = 70\) and \(Q_{1} = 30\). The pairs 40 and 60 and 20 and 80 add to 100 but give differences of 20 and 60, so skewness of 1.2 and 0.4. The pair 30 and 60 does not even add to 100.

Answer: B

The geometric mean of 4, 8 and 16 is 8.0. It is the cube root of the product of the three numbers.

\[ GM = (4 \times 8 \times 16)^{1/3} \]

\[ = (512)^{1/3} = 8 \]

The arithmetic mean would be 9.33, which is option A and a common slip. The values 4.8 and 10.2 do not come from either mean. The answer is option B.

A useful check is that the three numbers form a geometric progression with ratio 2, and the middle term of such a progression is its geometric mean.