 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/16/h/solution

The requested bounds are

E [X] - σ = 2/5

and

E [X] + σ = 4/5

, both within the support. Since the probability distribution is continuous, endpoints have zero probability, and subtraction of the cumulative distribution function gives

P (E [X] - σ \leq X \leq E [X] + σ) = 4 [(\frac{4}{5})^{3} - (\frac{2}{5})^{3}] - 3 [(\frac{4}{5})^{4} - (\frac{2}{5})^{4}] .

This leaves the answer in rational fractions as requested.

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