Unlike discrete variables which are counted (e.g., number of cracks), continuous variables are measured (e.g., length of a beam, curing time of concrete, water pressure, compressive strength). Because a continuous variable can take on infinitely many values within a range, the probability of it taking any specific exact value is practically zero. Instead, engineers calculate the probability that the variable falls within a specified interval, such as the probability that concrete strength is between 25 MPa and 30 MPa.

The continuous analog to a probability mass function.

A valid Probability Density Function must satisfy two conditions:

• $f(x) \ge 0$ for all $x$.
• $\int_{-\infty}^{\infty} f(x) \, dx = 1$ (the total area under the curve is 1).

The Normal (Gaussian) distribution is pervasive because many natural phenomena, manufacturing processes, and measurement errors follow a bell-shaped curve. It is the most important continuous distribution in statistics. In civil engineering, the compressive strength of structural steel or the density of compacted soil often follows a normal distribution.

• It is centered at $\mu$, which is also its median and mode.
• The spread is determined by $\sigma$; a larger $\sigma$ results in a flatter, wider curve.

Because every normal distribution has a different mean and standard deviation, we convert any normal distribution to a standard scale to easily calculate probabilities. We standardize the variable $X$ into a $Z$-score, which represents the number of standard deviations $X$ is from the mean.

Once $Z$ is found, probabilities $P(Z \le z)$ are looked up in a Standard Normal Table (Z-table) or calculated via software.

Models used for specific engineering scenarios, particularly in reliability and failure analysis.

• Mean: $\mu = \frac{a+b}{2}$
• Variance: $\sigma^2 = \frac{(b-a)^2}{12}$

The Exponential distribution models the time between occurrences. In civil engineering, this could be the time between major earthquakes, the distance between severe potholes on a highway, or the lifespan of certain electronic sensors before failure.

• Mean: $\mu = \frac{1}{\lambda}$
• Variance: $\sigma^2 = \frac{1}{\lambda^2}$

Memoryless Property: The probability of failure in the next instant does not depend on how long the component has already survived. This makes it less suitable for modeling materials that undergo wear and tear (like fatigue).

These are critical distributions for reliability engineering and material strength.

Unlike the memoryless Exponential distribution, Weibull can model failure rates that increase over time (wear-out) or decrease over time (early infant mortality).

• $\beta$ (Shape parameter): If $\beta > 1$, the failure rate increases over time (wear-out phase).
• $\eta$ (Scale parameter or characteristic life): The time by which 63.2% of the population will have failed.