Exam PRandom variablesFree to read

Random variables, CDFs and survival functions

The CDF is the one object every distribution has; densities, masses, survival functions and percentiles are all read off it.

The formulas

CDF

$$F(x)=P(X\le x)$$

Interval probability

$$P(a<X\le b)=F(b)-F(a)$$

Survival function

$$S(x)=1-F(x)=P(X>x)$$

Density

$$f(x)=F^{\prime }(x)$$

continuous case

Percentile

$$F({\pi }_p)=p$$

Hazard rate

$$h(x)=\frac{f(x)}{S(x)}=-\frac{d}{dx}\ln S(x)$$

Where it comes from

1. $F$ is non-decreasing, right-continuous, and runs from 0 to 1 — those three properties characterise a valid CDF.
2. $P(a<X\le b)=F(b)-F(a)$ follows from additivity on the disjoint pieces $(-\infty ,a]$ and $(a,b]$.
3. For a continuous variable the endpoints carry no probability, so $<$ and $\le$ can be used interchangeably; for a discrete one they cannot.

Worked example

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

• Everything is F(b) − F(a). If you can write the CDF, you can answer the question.
• S(x) = 1 − F(x), and for the exponential S(x) = e^(−λx) is the form worth memorising directly.

Traps

• Using < and ≤ interchangeably for a DISCRETE variable, where the endpoint carries real mass.
• Differentiating a CDF that has a jump — a mixed distribution needs the jump handled separately.

Related

• Expectation, variance and moments
• Transformations and order statistics
• Exponential distribution

Drill this: the Exam P question bank has original questions on this topic, and today’s free round is open to everyone.