### II. Probability and Statistics for Reliability

A. Basic concepts

## 3. Discrete and continuous probability distributions (Analyze)

*Compare and contrast various distributions (binomial, Poisson, exponential, Weibull, normal, log-normal, etc.) and their functions (e.g., cumulative distribution functions (CDFs), probability density functions (PDFs), hazard functions), and relate them to the bathtub curve.*

This lesson takes a close look at the continuous distributions commonly used in reliability engineering.

## Additional References

Interpolation within Distribution Tables (article)

Reading a Standard Normal Table (article)

The Normal Distribution (article)

Lognormal Distribution (article)

Calculating Lognormal Distribution Parameters (article)

The Exponential Distribution (article)

Using The Exponential Distribution Reliability Function (article)

Weibull Distribution (article)

Calculate Weibull Mean and Variance (article)

## Quick Quiz

1-18. Which distribution is used to describe the time between failures that occur independently at a constant rate?

(A) exponential

(B) gamma

(C) lognormal

(D) Weibull

1-22. What is the approximate reliability at the mean time to failure for the exponential model?

(A) 34%

(B) 37%

(B) 50%

(C) 67%

1-27. Consider a Weibull distribution. What is the scale parameter, as a characteristic of time to failure, as a percentile of the distribution?

(A) 31.6

(B) 36.7

(C) 63.2

(D) 63.3

1-28. A test shows four failures in 40 hours of operation. If the failure rate is constant, how many failures will the test show in 800 hours of operation?

(A) 4

(B) 8

(C) 80

(D) 160

1-29. A trans-African safari is to be made using a special custom-made four-wheeled vehicle equipped with five tires. The probability of failure for each tire on the safari follows a binomial distribution and is estimated to be 0.4. Calculate the probability that the safari can be completed successfully with the five available tires?

(A) 0.1296

(B) 0.2592

(C) 0.3370

(D) 0.4752

1-30. An earthquake prediction network has been determined to have a mean time to failure of a constant 130 hours. Calculate its reliability at t = 135 hours?

(A) 0.354

(B) 0.368

(C) 0.632

(D) 0.646

1-36. Which of the following probability distributions is continuous?

(A) binomial

(B) hypergeometric

(C) Poisson

(D) Weibull

1-43. If Z is a continuous random variable with a density distribution of 1 ≤ Z ≤ 5, what is the probability that Z = 4.0?

(A) 0.00

(B) 0.20

(C) 0.30

(D) 0.40

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