2012/01/03 by Charles Hoffman, Hoffman, Charles, Corey Manack +1
Mathematics · #05A15 #05A19 #60G50 #Combinatorics (math.CO) #FOS: Mathematics #Statistics Theory (math.ST) #math.CO #math.ST #msc:05A15 #msc:05A19 #msc:60G50 #stat.TH
paper · pdf · doi:10.48550/arxiv.1201.0571
10 pages, 2 figures
arxiv created 2015/08/20 · arxiv updated 2015/08/21
We enumerate the number of monotonic lattice paths starting at (0,0) and terminating at (m,n) in which l of the first k steps lie below the line y=x (0≤ k≤ m≤ n). These closed formulas consist of terms which are a product Catalan numbers, ballot numbers and binomial coefficients. We then apply the combinatorial formulas to failure analysis by deriving a probability distribution that compares the performance of a k-out-of-m system to a k-out-of-n system of continuous, independent, and identically distributed random variables. Lastly, we provide asymptotics in a few special cases of k,m,n and leave others as conjecture.