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QLDMath MethodsQuick questions
Unit 4: Further calculus and statistical inference
Quick questions on Continuous random variables and pdf: QCE Maths Methods Unit 4
10short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is finding a normalising constant?Show answer
A common PSMT and EA question gives $f(x) = k g(x)$ on $[a, b]$ and asks for $k$. Set $\int k g = 1$; solve for $k$.
What is find $k$?Show answer
$\int_{0}^{2} k x \, dx = [\frac{k x^2}{2}]_{0}^{2} = 2k = 1$, so $k = 1/2$.
What is find $E $?Show answer
$E(X) = \int_{0}^{2} x \cdot \frac{x}{2} \, dx = \int_{0}^{2} \frac{x^2}{2} \, dx = \frac{8}{6} = \frac{4}{3}$.
What is find $\text{Var} $?Show answer
$\text{Var}(X) = 2 - (4/3)^2 = 2 - 16/9 = (18-16)/9 = 2/9$.
What is find SD?Show answer
$\sigma = \sqrt{2/9} = \sqrt{2}/3$.
What is forgetting normalisation?Show answer
A pdf must integrate to 1. Forgetting this in finding $k$ is the most common slip.
What is using $f $ as probability?Show answer
$f(2) = 0.3$ does not mean $P(X = 2) = 0.3$. The pdf is a density, not a probability.
What is wrong variance formula?Show answer
$\text{Var} = E(X^2) - [E(X)]^2$, not $E(X^2) - E(X)$.
What is forgetting the pdf inside $E $?Show answer
$E(X) = \int x f(x) \, dx$, not $\int x \, dx$.
What is wrong support?Show answer
If $f = 0$ outside $[a, b]$, the integrals must be on $[a, b]$, not $(-\infty, \infty)$.