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NSWMaths AdvancedQuick questions
Year 12: Statistical Analysis
Quick questions on Continuous random variables: probability density functions, cumulative distributions, mean and variance
13short 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 probability density functions?Show answer
A continuous random variable $X$ is described by a probability density function $f$ satisfying
What is probabilities as integrals?Show answer
For any interval $[c, d]$ inside the support,
What is cumulative distribution function?Show answer
The cumulative distribution function (cdf) is
What is mean, variance, median, mode?Show answer
$$E(X) = \mu = \int_{-\infty}^{\infty} x f(x) \, dx.$$
What is finding a constant and a probability?Show answer
$f(x) = c(1 - x^2)$ for $-1 \le x \le 1$, $0$ elsewhere. Find $c$, then $P(X > 0)$.
What is computing the mean and variance?Show answer
For $f(x) = \frac{x}{8}$ on $[0, 4]$:
What is cdf from a pdf?Show answer
For $f(x) = \frac{x}{8}$ on $[0, 4]$, the cdf is
What is median and mode?Show answer
For $f(x) = \frac{x}{8}$ on $[0, 4]$, the median $m$ solves $\frac{m^2}{16} = \frac{1}{2}$, so $m^2 = 8$ and $m = 2 \sqrt{2} \approx 2.83$.
What is treating $P > 0$?Show answer
For a continuous random variable, single points have zero probability. The pdf value $f(c)$ is a density, not a probability.
What is forgetting to enforce total probability?Show answer
When a pdf has an unknown constant, the first step is always $\int f = 1$.
What is confusing pdf and cdf?Show answer
$f$ is the density (can exceed $1$), $F$ is the cumulative probability (always between $0$ and $1$). $f = F'$ where $F$ is differentiable.
What is integrating $x f $ over the wrong range?Show answer
When computing $E(X)$, integrate only over the support. Outside the support, $f = 0$ contributes nothing.
What is picking an interior critical point that is actually a minimum?Show answer
The mode is the maximum of $f$. If $f$ is monotone, the mode is at an endpoint of the support.