Probability describes a population before data are observed. Statistics uses a random sample to learn about that population. Keeping these directions separate prevents many later confusions.
1. Outcomes, events, and random variables
An outcome is a possible result of a random process; the sample space \(\Omega\) contains all outcomes. An event A is a subset of \(\Omega . A\) random variable \(Y\) maps an outcome to a numerical value. Probabilities are nonnegative\(, \Pr(\Omega )=1\), and probabilities of mutually exclusive events add.
A discrete variable takes a finite or countable set of values. A continuous variable with a density takes values on a continuum. Uppercase \(Y\) denotes the random variable; lowercase \(y\) denotes a possible or realized value. A parameter such as \(\mu\) describes its population distribution.
2. PMF, CDF, and the Bernoulli distribution
The probability mass function (PMF) is \(p_{Y}(y)=\Pr(Y=y)\). For support \({y_{1},\ldots ,y_{k}}\), each probability is nonnegative and their sum is one. Event probabilities are sums over the relevant support points.
The cumulative distribution function (CDF) is nondecreasing and right-continuous, with limits zero and one at the two extremes. For any \(a<b, \Pr(a<Y\le b)=F_{Y}(b)-F_{Y}(a)\). At a discrete mass point\(, \Pr(Y=a)=F_{Y}(a)-F_{Y}(a-)\); endpoint inclusion matters.
| Number of crashes M | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \(\Pr(M=m)\) | .80 | .10 | .06 | .03 | .01 |
| \(F_{M}(m)\) | .80 | .90 | .96 | .99 | 1 |
Thus \(\Pr(1\le M\le 2)=.10+.06=.16\), whereas \(\Pr(1<M\le 2)=.06\). The CDF is a staircase, with jump sizes equal to the PMF.
A Bernoulli variable equals 1 with probability \(p\) and 0 with probability \(1-p\). It encodes any binary outcome: employment, voting, treatment assignment, or a short commute. Its parameter is a probability, not an observed sample fraction.
3. Continuous distributions and density
For a variable with a density\(, \Pr(Y=c)=0\). The density is a height, not a point probability, and can exceed one. For example, the uniform density on [0,.5] has height 2; the total area is still one. Where F is differentiable\(, f=F'\).
Worked example. If the commuting-time CDF satisfies \(F(15)=.20\) and \(F(20)=.78\), then \(\Pr(15<Y\le 20)=.58\) and \(\Pr(Y>20)=.22\). Both a CDF difference and a density area describe the same event.
4. Quantiles and distribution reading
The \(\tau\)-quantile is \(q_\tau=\inf\{y:F(y)\ge\tau\}\). The median is \(q_{.5}\); the interquartile range is \(q_{.75}-q_{.25}\). For continuous strictly increasing CDFs, \(F(q_\tau)=\tau\). The generalized definition also works for discrete variables.
Percentiles become critical values in inference: the 97.5th percentile of a standard normal distribution leaves 2.5% in its upper tail. A median describes a distribution'\(s\) location but need not equal its mean.
Check your understanding
Question. A density equals 3 on an interval of width .1. What is the probability of that interval? Can a discrete PMF equal 3?
Solution. The probability is .3. A PMF cannot exceed 1, because each value is an event probability.
Sources: SW §2.1, printed pp. 61–65 (PDF pp. 2–6); percentile interpretation in Table 2.4, p. 80. Quantile definition and density calculus are supporting extensions. Next: moments turn distributions into useful numerical summaries.