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VICMath MethodsQuick questions
Unit 4
Quick questions on Continuous random variables: VCE Math Methods Unit 4
11short 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 typical Paper 2 question gives $f(x) = k g(x)$ on some interval and asks for the value of $k$ that makes $f$ a valid pdf.
What is linearity of expectation?Show answer
$E(aX + b) = a E(X) + b$ for constants $a, b$.
What is expectation of a function?Show answer
$$E[g(X)] = \int_{a}^{b} g(x) f(x) \, dx$$
What is properties?Show answer
$\text{Var}(aX + b) = a^2 \text{Var}(X)$. Adding a constant does not change variance; multiplying by a constant scales variance by the constant squared.
What is worked variance?Show answer
For the pdf $f(x) = \frac{x}{8}$ on $[0, 4]$ (worked above), $E(X) = \frac{8}{3}$, $E(X^2) = 8$, $\text{Var}(X) = \frac{8}{9}$, $\sigma = \frac{2 \sqrt{2}}{3}$.
What is forgetting the normalisation condition?Show answer
A pdf must integrate to 1 over its support. Forgetting to find $k$ from this condition is the most common Paper 2 mistake.
What is using $f $ as if it were a probability?Show answer
$f(2) = 0.3$ does not mean "$P(X = 2) = 0.3$". For a continuous random variable, $P(X = 2) = 0$ always. The pdf is a density, not a probability.
What is wrong formula for variance?Show answer
$\text{Var}(X) = E(X^2) - [E(X)]^2$, not $E(X^2) - E(X)$ or $E(X)^2 - E(X^2)$. The square applies to the mean.
What is forgetting to use the pdf inside $E $?Show answer
$E(X) = \int x f(x) \, dx$, not $\int x \, dx$. The pdf is the weight.
What is treating support as $ $ when it is $[a, b]$?Show answer
If $f$ is zero outside $[a, b]$, do not integrate outside $[a, b]$; the integral is zero there. Set limits to $a$ and $b$.
What is confusing median and mean?Show answer
For skewed pdfs the median and mean differ. Symmetric pdfs have median equal to mean.