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NSWMaths AdvancedQuick questions
Year 12: Statistical Analysis
Quick questions on Discrete random variables: probability distribution, expected value, variance and standard deviation
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 discrete random variables and their distributions?Show answer
A discrete random variable $X$ takes a countable list of values $x_1, x_2, \dots, x_n$ with probabilities $p_i = P(X = x_i)$. The list of values with their probabilities is the probability distribution of $X$. For it to be valid:
What is expected value?Show answer
The expected value (or mean) of $X$ is the long-run average value if we repeated the experiment many times. It is the weighted sum
What is expected value of a function of $X$?Show answer
$$E(g(X)) = \sum_i g(x_i) \, p_i.$$
What is variance and standard deviation?Show answer
The variance of $X$ measures spread around the mean. It is
What is linear transformations?Show answer
If $Y = a X + b$ for constants $a$ and $b$,
What is checking a distribution and computing the mean?Show answer
$X$ takes values $1, 2, 3, 4$ with $P(X = x) = c x$ for some constant $c$. Find $c$, then $E(X)$.
What is variance via $E(X^2) - \mu^2$?Show answer
With the same $X$, $E(X^2) = 1(0.1) + 4(0.2) + 9(0.3) + 16(0.4) = 0.1 + 0.8 + 2.7 + 6.4 = 10$.
What is a fair die?Show answer
$X$ is the number rolled on a fair six-sided die. $E(X) = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5$.
What is linear transformation?Show answer
If $X$ has $\mu = 3.5$ and $\sigma^2 = \frac{35}{12}$, then $Y = 2 X + 1$ has $E(Y) = 2(3.5) + 1 = 8$ and $\text{Var}(Y) = 4 \cdot \frac{35}{12} = \frac{35}{3}$.
What is forgetting to check that probabilities sum to $1$?Show answer
If a question gives a distribution in terms of a constant, solve $\sum p_i = 1$ first.
What is using $\mu^2$ instead of $E $?Show answer
The formula is $\text{Var}(X) = E(X^2) - [E(X)]^2$, not $E(X^2) - E(X)$.
What is squaring inside but not outside?Show answer
$E(X)^2 = \mu^2$ is the square of a single number. $E(X^2) = \sum x^2 p$ is the weighted sum of squares. They are different.
What is linear transformation on variance?Show answer
$\text{Var}(a X + b) = a^2 \text{Var}(X)$, not $a \text{Var}(X)$, and the $+ b$ has no effect on variance.
What is negative variance?Show answer
If you get a negative variance, you have a calculation error. Variance is always non-negative.