 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/discrete-mathematics-probability-and-continuous-mathematics/9/a/ii/solution

The factor

2 x - x^{3} = x (2 - x^{2})

is positive on

(0, 2)

and negative on

(2, 2)

. For every nonzero

α

, the proposed probability density function is therefore negative on an interval. The choice

α = 0

has total integral zero. In fact,

\int_{0}^{2} (2 x - x^{3}) d x = [x^{2} - \frac{x ^{4}}{4}]_{0}^{2} = 0,

so no scaling could normalise it even without the sign obstruction. Consequently

No value of α gives a probability density function; no corresponding expectation exists .

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