A theoretical justification based on a certain material failure mechanism underlies the assumption that ductile strength(延展强度) $X$ of a material has a lognormal distribution.

[基於某種材料失效機制的理論論證是以假定材料的延展強度 $X$ 具有對數正態分佈為基礎。]

Suppose the parameters are $\mu=5$ and $\sigma=.1$.

• (a) Compute $E(X)$ and $V(X)$.
• (b) Compute $P(X > 120)$.
• (c) Compute $P(110\leqslant X\leqslant 130)$.
• (d) What is the value of median ductile strength?
• (e) If ten different samples of an alloy steel(合金鋼) of this type were subjected to a strength test, how many would you expect to have strength at least $120$?
• (f) If the smallest $5\%$ of strength values were unacceptable, what would the minimum acceptable strength be?