Probability, Random Variables, PDFs, and CDFs | Free Exam P Lesson

The actuary reads $F (70) = 0.70$ and prices as though 70% die before age 70. The survival function is $S (70) = 1 - F (70) = 0.30$ . Swapping CDF and survival flips the answer.

HIGH-FREQUENCY: The relationship between PDF, CDF, and survival function (and moving fluidly between them) appears on nearly every continuous-variable problem.

A random variable $X$ maps outcomes in a sample space to real numbers.

Discrete $X$ : takes values in a countable set. Described by a PMF $p_{X} (x) = P (X = x)$ , where $\sum p_{X} (x) = 1$ .

Continuous $X$ : takes values in an interval. Described by a PDF $f_{X} (x) \geq 0$ , where:

$\int_{- \infty}^{\infty} f_{X} (x) d x = 1$

Critical distinction: for continuous variables, $P (X = x) = 0$ for every point. Probabilities come only from integrating:

$P (a \leq X \leq b) = \int_{a}^{b} f_{X} (x) d x$

So $P (a < X < b) = P (a \leq X \leq b)$ , endpoints do not matter for continuous variables.

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Common mistakes

Confusing CDF with survival. Exponential with mean 100: $F (50) = 1 - e^{- 0.5} = 0.3935$ , $S (50) = e^{- 0.5} = 0.6065$ . Reading the wrong one flips the answer. Trap: 0.6065 when the answer is 0.3935.
Forgetting $P (X = x) = 0$ for continuous variables. $P (X \geq 2) = P (X > 2)$ when $X$ is continuous. For discrete, $P (X \geq 2) \neq = P (X > 2)$ . Trap: subtracting a nonexistent point mass.
Not normalizing a piecewise density. Setting $c = 1$ in $f (x) = c x^{2}$ on $(0, 3)$ gives $P (1 < X < 2) = 7/3 > 1$ . Trap: probability exceeding 1.

Bottom line

CDF: $F_{X} (x) = P (X \leq x)$ ; survival $S_{X} (x) = 1 - F_{X} (x)$ . Confirm which one the question wants
Continuous variables: $P (X = x) = 0$ , so probabilities come from integrating the density
PDF $\leftrightarrow$ CDF: integrate the density to get the CDF, differentiate the CDF to get the density
Unknown constant: integrate the density over its support and set the result equal to 1

Exam shortcut

When you see a piecewise density with unknown constant, find the constant first, integrate over the full support and set equal to 1. Then compute whatever is asked. "CDF goes up, survival goes down." CDF starts at 0, ends at 1. Survival starts at 1, ends at 0. If your survival exceeds 1 or your CDF decreases, you have an error.

The full lesson (about 2,381 words, 16 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

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