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Date November 2012 Marks available 3 Reference code 12N.1.hl.TZ0.5
Level HL only Paper 1 Time zone TZ0
Command term Determine Question number 5 Adapted from N/A

Question

The continuous random variable X has probability density function given by

\[f(x) = \left\{ {\begin{array}{*{20}{c}}
  {a{e^{ - x}},}&{0 \leqslant x \leqslant 1} \\
  {0,}&{{\text{otherwise}}{\text{.}}}
\end{array}} \right.\]

State the mode of X .

[1]
a.

Determine the value of a .

[3]
b.

Find E(X ) .

[4]
c.

Markscheme

0     A1

[1 mark]

a.

\(\int_0^1 {f(x)dx = 1} \)     (M1)

\( \Rightarrow a = \frac{1}{{\int_0^1 {{e^{ - x}}dx} }}\)

\( \Rightarrow a = \frac{1}{{\left[ { - {e^{ - x}}} \right]_0^1}}\)

\( \Rightarrow a = \frac{e}{{e - 1}}\) (or equivalent)     A1

Note: Award first A1 for correct integration of \(\int {{e^{ - x}}dx} \) .

This A1 is independent of previous M mark.

 

[3 marks]

b.

\({\text{E}}(X) = \int_0^1 {xf(x)dx\left( { = a\int_0^1 {x{e^{ - x}}dx} } \right)} \)     M1

attempt to integrate by parts     M1

\( = a\left[ { - x{e^{ - x}} - {e^{ - x}}} \right]_0^1\)     (A1)

\( = a\left( {\frac{{e - 2}}{e}} \right)\)

\( = \frac{{e - 2}}{{e - 1}}\) (or equivalent)     A1

[4 marks]

c.

Examiners report

A range of answers were seen to part a), though many more could have gained the mark had they taken time to understand the shape of the function. Part b) was done well, as was part c). In c), a number of candidates integrated by parts, but found the incorrect expression \( - x{e^{ - x}} + {e^{ - x}}\).

a.

A range of answers were seen to part a), though many more could have gained the mark had they taken time to understand the shape of the function. Part b) was done well, as was part c). In c), a number of candidates integrated by parts, but found the incorrect expression \( - x{e^{ - x}} + {e^{ - x}}\).

b.

A range of answers were seen to part a), though many more could have gained the mark had they taken time to understand the shape of the function. Part b) was done well, as was part c). In c), a number of candidates integrated by parts, but found the incorrect expression \( - x{e^{ - x}} + {e^{ - x}}\).

c.

Syllabus sections

Topic 5 - Core: Statistics and probability » 5.5 » Definition and use of probability density functions.
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