S2 June 2011 Q3

EdexcelOld spec10 marksContinuous Random Variables

3.

Figure 1: sketch of f(x): a curve rising from O to the point B above x = 3, then a straight line down to the x-axis at a
Figure 1

Figure 1 shows a sketch of the probability density function \(\mathrm{f}(x)\) of the random variable \(X\).

For \(0 \leqslant x \leqslant 3\), \(\mathrm{f}(x)\) is represented by a curve \(OB\) with equation \(\mathrm{f}(x) = kx^2\), where \(k\) is a constant.

For \(3 \leqslant x \leqslant a\), where \(a\) is a constant, \(\mathrm{f}(x)\) is represented by a straight line passing through \(B\) and the point \((a, 0)\).

For all other values of \(x\), \(\mathrm{f}(x) = 0\).

Given that the mode of \(X\) = the median of \(X\), find

(a) the mode, (1)
(b) the value of \(k\), (4)
(c) the value of \(a\). (3)

Without calculating \(\mathrm{E}(X)\) and with reference to the skewness of the distribution

(d) state, giving your reason, whether \(\mathrm{E}(X) \lt 3\), \(\mathrm{E}(X) = 3\) or \(\mathrm{E}(X) \gt 3\). (2)
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