Continuous Probability Distributions - Examples & Applications

The compressive strength of concrete samples is normally distributed with a mean $\mu = 4000$ psi and a standard deviation $\sigma = 200$ psi. What is the probability that a randomly selected sample has a strength less than $3800$ psi?

Using the same parameters as Problem 1 ($\mu = 4000$ psi, $\sigma = 200$ psi), what is the probability that a sample's strength falls between $3900$ psi and $4300$ psi?

A steel manufacturing process produces rebars with a yield strength that is normally distributed with a standard deviation $\sigma = 15.0$ MPa. If the manufacturer wants to ensure that $95.0\%$ of the rebars have a yield strength greater than $400$ MPa, what must be the target mean yield strength $\mu$?

The fatigue life $N$ (in millions of cycles) of a steel bridge component is modeled using a lognormal distribution. The parameters of the corresponding normal distribution for $\ln(N)$ are $\mu = 2.500$ and $\sigma = 0.4000$. Determine the probability that the component will fail before reaching $10.0$ million cycles.

The time between breakdowns of an earthmoving excavator follows an exponential distribution with a mean time between failures $\theta = 500$ hours. What is the probability that the excavator will operate for at least $600$ hours without breaking down?

Using the same excavator from Problem 5 ($\lambda = 0.002000$ failures/hour), suppose the machine has already operated successfully for $400$ hours. What is the probability that it will operate for an additional $600$ hours without failure?

A specialized surveying drone is scheduled to arrive at a construction site between 8:00 AM and 10:00 AM. Its arrival time is uniformly distributed over this interval. If an engineer starts waiting at 8:30 AM, what is the probability they will have to wait more than $45.0$ minutes for the drone to arrive?

For the surveying drone in Problem 7 with arrival times uniformly distributed between $0$ and $120$ minutes past 8:00 AM, calculate the expected arrival time and the standard deviation of the arrival time.

A pump system in a wastewater treatment plant requires three independent bearing failures to completely halt operation. The time to failure for a single bearing follows an exponential distribution with a mean of $400$ days. What is the probability that the entire pump system fails before $1000$ days?

The maximum annual wind speed at a bridge site is modeled by a Weibull distribution with a shape parameter $\beta = 2.000$ (Rayleigh distribution) and a scale parameter $\eta = 25.0$ m/s. What is the probability that the maximum wind speed in a given year exceeds $35.0$ m/s?

A structural cable is known to have a breaking strength governed by a Weibull distribution with shape parameter $\beta = 5.000$ and scale parameter $\eta = 800$ kN. A design engineer needs to find the "B10 life" or the characteristic load below which only $10.0\%$ of the cables will fail.

In a construction project, the duration of a critical foundation-laying activity is modeled using a Beta-PERT distribution. The optimistic time is $a = 10.0$ days, the most likely time is $m = 14.0$ days, and the pessimistic time is $b = 24.0$ days. Calculate the expected duration and the standard deviation of this activity.