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: $f(x) = F'(x)$
continuous case
Percentile: $F(\pi_p) = p$
Hazard rate: $h(x) = \dfrac{f(x)}{S(x)} = -\dfrac{d}{dx}\ln S(x)$

Where it comes from

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

Worked example

A loss has CDF $F(x) = 1 - (1+x)^{-3}$ for $x > 0$ (in thousands). Find the probability the loss is between 1,000 and 4,000.

$P(1 < X \le 4) = F(4) - F(1)$ - always take the difference of the CDF, never integrate twice.
$F(4) = 1 - 5^{-3} = 1 - 0.008 = 0.992$ .
$F(1) = 1 - 2^{-3} = 1 - 0.125 = 0.875$ .
$P = 0.992 - 0.875 = 0.117$ .

Answer: 0.1170

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.

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Random variables, CDFs and survival functions

The formulas

Where it comes from

Worked example

Memory hooks

Traps

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