| Date | May 2012 | Marks available | 3 | Reference code | 12M.2.hl.TZ2.7 |
| Level | HL only | Paper | 2 | Time zone | TZ2 |
| Command term | Find | Question number | 7 | Adapted from | N/A |
Question
The probability density function of a continuous random variable X is given by
\(f(x) = \frac{1}{{1 + {x^4}}}\), \(0\) \(''\) \(x\) \(''\) \(a\) .
Find the value of a .
Find the mean of X .
Markscheme
\(\int_0^a {\frac{1}{{1 + {x^4}}}{\text{d}}x = 1} \) M2
a = 1.40 A1
[3 marks]
\({\text{E}}(X) = \int_0^a {\frac{x}{{1 + {x^4}}}{\text{d}}x} \) M1
\(\left( { = \frac{1}{2}\arctan ({a^2})} \right)\)
= 0.548 A1
[2 marks]
Examiners report
Many candidates picked up some marks for this question, but only a few gained full marks. In part (a) many candidates did not appreciate the need for the calculator to find a value of a. Candidates had more success with part (b) with a number of candidates picking up follow through marks.
Many candidates picked up some marks for this question, but only a few gained full marks. In part (a) many candidates did not appreciate the need for the calculator to find a value of a. Candidates had more success with part (b) with a number of candidates picking up follow through marks.
