Continuous Random Variable: Normalization and Expected Value

00:00 6 marks

Let \(X\) be a continuous random variable with probability density function
\[
f(x) = \begin{cases}
kx(4-x^2) & 0 \le x \le 2 \\
0 & \text{otherwise}
\end{cases}
\]
where \(k\) is a positive constant.

(a) Show that \(k = \frac{1}{4}\). [3]
(b) Find \(E(X)\). [3]

Part a

5 minutes 3 marks

Question

Show that \(k = \frac{1}{4}\). [3]

Hint Hint

Hint

Integrate \(kx(4-x^2)\) from 0 to 2, set the result equal to 1 and then solve for k.

Answer Answer
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Complete Answer

\(k = \frac{1}{4}\)

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Part b

5 minutes 3 marks

Question

Find \(E(X)\). [3]

Hint Hint

Hint

Substitute \(k = \frac{1}{4}\) into the expected value formula \(E(X)=\int_{0}^{2} x f(x)dx\) and evaluate the resulting integral.

Answer Answer
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Explanation Explanation

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Complete Answer

\(E(X)=\frac{16}{15}\)

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