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QLDMath MethodsQuick questions
Unit 3: Further calculus and statistics
Quick questions on Discrete random variables, expected value and variance (QCE Mathematical Methods Unit 3)
14short 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 random variables?Show answer
A random variable $X$ assigns a numerical value to each outcome of a probability experiment. A random variable is discrete if its possible values form a countable set (typically a list of integers).
What is expected value?Show answer
The expected value (or mean) of $X$ is the probability-weighted average of its values:
What is variance and standard deviation?Show answer
The variance measures spread around the mean.
What is the Bernoulli distribution?Show answer
A Bernoulli trial has exactly two outcomes, labelled success ($X = 1$) and failure ($X = 0$), with probability $p$ of success.
What is validity check?Show answer
A student claims the following is a probability distribution: $P(X = 1) = 0.2$, $P(X = 2) = 0.5$, $P(X = 3) = 0.4$. Is it valid?
What is mean and variance from a table?Show answer
A discrete random variable $X$ has $P(X = 1) = 0.4$, $P(X = 2) = 0.4$, $P(X = 5) = 0.2$.
What is linear transformation?Show answer
If $E(X) = 2.2$ and $\mathrm{Var}(X) = 2.16$, find $E(3 X + 1)$ and $\mathrm{Var}(3 X + 1)$.
What is bernoulli with $p = 0.4$?Show answer
$X \sim \mathrm{Bern}(0.4)$ has $E(X) = 0.4$ and $\mathrm{Var}(X) = 0.4 \cdot 0.6 = 0.24$.
What is forgetting the normalisation check?Show answer
A discrete distribution must have probabilities that sum to exactly 1. Forgetting to set this up costs the first mark when QCAA gives an unknown $k$.
What is confusing $E $ and $[E ]^2$?Show answer
The variance shortcut is $E(X^2) - [E(X)]^2$, where the first term squares each $x$ before weighting and the second squares after weighting. They are not equal.
What is dropping the square on the scale factor?Show answer
$\mathrm{Var}(a X + b) = a^2 \, \mathrm{Var}(X)$, not $a \, \mathrm{Var}(X)$.
What is computing standard deviation but stopping at variance?Show answer
If QCAA asks for standard deviation, take the square root.
What is naming Bernoulli when you mean binomial?Show answer
A Bernoulli is a single trial. A binomial is $n$ trials. Mixing the two is a frequent IA2 short response error.
What is using negative probabilities?Show answer
A probability cannot be negative. If your algebra produces a negative $p$, you have set up the problem wrongly.