Continuous Random Variable: PDF and Expected Value

00:00 5 marks

The continuous random variable $X$ has probability density function $f(x)$ given by

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

(a) Find the value of $k$. [2]
(b) Find $\text{E}(X)$. [3]

Part a

4 minutes 2 marks

Question

Find the value of $k$ such that $f(x)$ is a probability density function.

Hint Hint

Hint

Set the normalization condition $\int_{0}^{\pi} k \sin x\,dx = 1$ and solve for $k$.

Answer Answer
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Explanation Explanation

Answer Answer

Complete Answer

$k = \frac{1}{2}$

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Part b

6 minutes 3 marks

Question

Find $\text{E}(X)$ for the given probability density function.

Hint Hint

Hint

Use $\text{E}(X)=\int_{0}^{\pi} x f(x)\,dx$ with $k = \frac{1}{2}$.

Answer Answer
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Explanation Explanation

Answer Answer

Complete Answer

$\text{E}(X) = \frac{\pi}{2}$

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Answer Explanation Answer Explanation