IB Maths AA HL 4.14 Random variables and probability density

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The continuous random variable X has a probability density function given by

f(x)=\left\{\begin{array}{cc} k \sin \left(\frac{\pi x}{6}\right), & 0 \leq x \leq 6 \\ 0, & \text { otherwise } \end{array}\right.

(a)

Find the value of k.

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attempt to equate integral to 1 (may appear later)
$k \int_{0}^{6} \sin \left(\frac{\pi x}{6}\right) \mathrm{d} x=1$
correct integral
$k\left[-\frac{6}{\pi} \cos \left(\frac{\pi x}{6}\right)\right]_{0}^{6}=1$
substituting limits
$-\frac{6}{\pi}(-1-1)=\frac{1}{k}$ $k=\frac{\pi}{12}$

(b)

By considering the graph of f write down

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(i)

the mean of X;

[ 1 ]

(ii)

the median of X;

[ 1 ]

(iii)

the mode of X.

[ 1 ]

(c)

Show that $P(0 \leq X \leq 2)=\frac{1}{4}$ .

[ 4 ]

$\frac{\pi}{12} \int_{0}^{2} \sin \left(\frac{\pi x}{6}\right) \mathrm{d} x$

=\frac{\pi}{12}\left[-\frac{6}{\pi} \cos \left(\frac{\pi x}{6}\right)\right]_{0}^{2}

Note: Accept without the $\frac{\pi}{12}$ at this stage if it is added later.

\begin{aligned} & \frac{\pi}{12}\left[-\frac{6}{\pi}\left(\cos \frac{\pi}{3}-1\right)\right] \\ & =\frac{1}{4} \end{aligned}

IB Maths AA HL Ahl 4 14 Hl Random Variables and Probability Density Questions

Exam points

IB Maths AA HL Ahl 4 14 Hl Random Variables and Probability Density Questions question 1

Question (a)

Question (b)

Question (i)

Question (ii)

Question (iii)

Question (c)