The probability distribution of a discrete random variable X is defined by

\({\text{P}}(X = x) = cx(5 - x),{\text{ }}x = {\text{1, 2, 3, 4}}\) .

(a)     Find the value of c.

(b)     Find E(X) .

(a)     Using \(\sum {{\text{P}}(X = x) = 1} \)     (M1)

\(4c + 6c + 6c + 4c = 1\,\,\,\,\,(20c = 1)\)     A1

\(c = \frac{1}{{20}}\,\,\,\,\,( = 0.05)\)     A1     N1

(b)     Using \({\text{E}}(X) = \sum {x{\text{P}}(X = x)} \)     (M1)

\( = (1 \times 0.2) + (2 \times 0.3) + (3 \times 0.3) + (4 \times 0.2)\)     (A1)

\( = 2.5\)     A1     N1

Notes: Only one of the first two marks can be implied.

Award M1A1A1 if the x values are averaged only if symmetry is explicitly mentioned.

[6 marks]

This question was generally well done, but a few candidates tried integration for part (b).