Statistical distributions

Explanation

• A discrete probability distribution lists possible values with probabilities between $0$ and $1$ whose total is $1$; a discrete uniform distribution assigns equal probability to each value. Use $X\sim\operatorname{B}(n,p)$ for a fixed number $n$ of independent trials, each with two outcomes and constant success probability $p$.
• For a binomial variable, $P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}$; cumulative probabilities are often most efficiently found with a calculator.
• Translate inequalities carefully: for integer-valued $X$, $P(X<r)=P(X\leq r-1)$ and $P(X\geq r)=1-P(X\leq r-1)$.
• Use the cumulative binomial probability $P(X\leq4)=\sum_{r=0}^{4}\binom{14}{r}(0.4)^r(0.6)^{14-r}=0.279256\ldots$.
• Therefore $P(X\leq4)=0.2793$.

Explanation

• Write $X\sim\operatorname{N}(\mu,\sigma^2)$, where $\mu$ is the mean and $\sigma$ is the standard deviation; standardise with $Z=\frac{X-\mu}{\sigma}$.
• The Normal curve is continuous, symmetric about $\mu$, has total area $1$, and has points of inflection at $\mu-\sigma$ and $\mu+\sigma$.
• A Normal model is plausible for a roughly symmetric, unimodal histogram with no strong outliers, but context and the variable's possible values also matter.
• A binomial distribution may be approximated by a Normal distribution when $n$ is large and $p$ is close to $0.5$; use a continuity correction.

Explanation

• Select a distribution by matching its assumptions to the variable and data-generating process, not merely because its parameters can be estimated.
• A binomial model needs a fixed number of trials, two outcomes per trial, independence and a constant success probability.
• A Normal model is continuous and symmetric with unbounded tails, so it can be unsuitable for strongly skewed, bounded or discrete data.
• Support a choice with contextual evidence such as histogram shape, stability over time and dependence; state how a failed assumption could affect predictions.