After a thorough investigation into the stress at the extreme fibers of a steel beam, there are some significant conclusions to note.

The first deals with the calculations for s.

• Upon an initial review of the histogram plot for s, a lognormal distribution seems apparent. After applying the chi-square test for goodness of fit, the lognormal distribution of s is confirmed.
• The yield stress exceedence probability analysis resulted in the relationship between P_f and the sample size, N, at a P_f value of 0.072. Another conclusion drawn from this analysis is that the COV converges with respect to N as well.

After performing the parametric analysis, the most significant conclusion is that M has the largest impact on the number of beam failures. The c and I values do not have nearly as large an impact.

Additional analysis using hypothesis testing showed that the mean of the sample fell within a reasonable range of what was given in the problem statement as well as the distribution types initially stated.

There is some chance for error in the investigation of s. The probabilistic characteristics of M, c, I, and fy all showed slight discrepancies. In the statistics recorded for M, the variance was large relative to the other variables and may have adversely affected the results. There is also some error in the plots for the parametric analysis. None of the graphs are completely linear, but the assumption was made that the trends were all linear.

Further analysis could be performed to increase the sample size and thus reduce the sample COV more and provide more accurate results. However, the results show that the random numbers generated accurately indicated beam failures and the associated computed statistics were within an acceptable limit.