Probability Density Function and Median

00:00 7 marks

The probability density function of a continuous random variable $X$ is given by

$$ f(x) = \begin{cases} k \sin(\pi x) & 0 \le x \le 1 \\ 0 & \text{otherwise} \end{cases} $$

(a) Find the exact value of $k$. [3]

(b) Find $\textrm{P}(X \le 0.25)$. [2]

3 minutes 3 marks

Find the exact value of $k$.

Hint Hint

Set $\int_0^1 k \sin(\pi x) dx = 1$ and solve for $k$.

$k = \frac{\pi}{2}$

3 minutes 2 marks

Find $\textrm{P}(X \le 0.25)$.

Hint Hint

Integrate $\frac{\pi}{2} \sin(\pi x)$ from 0 to 0.25 and simplify.

$\textrm{P}(X \le 0.25) = \frac{1}{2}\left(1 – \frac{\sqrt{2}}{2}\right) = \frac{2-\sqrt{2}}{4}$ (approximately 0.146)

3 minutes 2 marks

Find the median of $X$.

Hint Hint

Set $\int_0^m \frac{\pi}{2} \sin(\pi x) dx = 0.5$ and solve for $m$.

The median is $m = 0.5$.