Continuous Univariate Distributions

Study guide: Exam P-1

A Continuous Univariate Distribution describes the probability law of a single random variable $X$ that can take any value in a continuous interval or union of continuous intervals.

The probability density function (PDF) $f(x)$ must satisfy $f(x) \geq 0$ and $\int_{-\infty}^{\infty} f(x)\,dx = 1$
Individual point probabilities are zero; all probabilities are computed as areas under $f$ :

$P(a \leq X \leq b) = \int_a^b f(x)\, dx$

The CDF and the density are two views of the same object: $F(x) = \int_{-\infty}^{x} f(t)\,dt$ and $f(x) = F'(x)$ . For anything phrased as a tail probability, work from the survival function $S(x) = 1 - F(x)$ instead.

Choosing the right one

Read the question for the story the quantity follows, then check the support matches:

Every value in a range equally likely → Uniform Continuous Distribution $(a,b)$ on $(a,b)$ , with $E[X] = (a+b)/2$
Waiting time between events, memoryless → Exponential Distribution $(\theta)$ on $(0,\infty)$ , with $E[X] = \theta$
Waiting time to the $\alpha$ -th event, or a skewed severity → Gamma $(\alpha,\theta)$ on $(0,\infty)$ , with $E[X] = \alpha\theta$
A proportion, rate, or probability → Beta $(\alpha,\beta)$ on $(0,1)$ , with $E[X] = \alpha/(\alpha+\beta)$
A sum or average of many variables → Normal Distribution $(\mu,\sigma^2)$ on $\mathbb{R}$ , with $E[X] = \mu$
A positive, heavily right-skewed loss whose log is normal → Lognormal Distribution $(\mu,\sigma^2)$ on $(0,\infty)$ , with $E[X] = e^{\mu+\sigma^2/2}$
The word memoryless is decisive: it appears only for the exponential (and its discrete counterpart, the geometric).
For the lognormal, $\mu$ and $\sigma$ are the parameters of $\ln X$ , not the mean and standard deviation of $X$ . Every lognormal probability reduces to a normal one by taking logs of both sides: $P(X > c) = P(\ln X > \ln c)$ .
Apart from a uniform with $a < 0$ , the normal is the only one on this list that can go negative, and the only one with no lower bound at all — a red flag if the quantity is a loss.
Insurance provisions (deductibles, limits, coinsurance) transform whichever severity distribution is chosen; see Transformations of Random Variables.

Finding a Probability from a PDF A random variable $X$ has PDF $f(x) = 3x^2$ for $0 < x < 1$ . Find $P(0.5 < X < 1)$ .

Answer $P(0.5 < X < 1) = \int_{0.5}^{1} 3x^2\, dx = \left[x^3\right]_{0.5}^{1} = 1 - (0.5)^3 = 1 - 0.125 = 0.875$