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The expected value, or mean, of a lognormal distribution is an important concept that helps us understand the central tendency of the data. For a variable \(X\) that is lognormally distributed, the expected value \(E(X)\) is determined using the formula \(E(X) = e^{\mu + \sigma^2/2}\), where \(\mu\) and \(\sigma\) are the parameters of the normal distribution from which the lognormal is derived.
In the exercise, \(\mu = 5\) and \(\sigma = 0.1\). By substituting these values into the formula, we calculate that \(E(X) = e^{5 + 0.1^2/2} = e^{5.005}\). This results in an expected value of approximately 148.41. This value represents the average ductile strength you would expect if you were to sample this material repeatedly.

Variance measures the spread or dispersion of a set of values in a distribution. For a lognormal distribution, the variance \(V(X)\) is calculated using the formula \(V(X) = \left(e^{\sigma^2} - 1\right) e^{2\mu + \sigma^2}\).
In our case, where \(\sigma = 0.1\) and \(\mu = 5\), this formula becomes \(V(X) = \left(e^{0.1^2} - 1\right) e^{10.01}\). This computation yields approximately 225.00. The variance is crucial because it provides us with a measure of how much variation from the average (or expected value) we can expect. A higher variance implies more spread out data, which might mean less predictable strength in material characteristics.

Probability calculations help us determine the likelihood of an event within a lognormal distribution. To calculate probabilities like \(P(X > 125)\), we can convert the problem to the standard normal distribution by using \(Z\)-scores.
First, we find the natural logarithm of 125, \(\ln(125) = 4.8283\). We use this to find the corresponding \(Z\)-score, \(\frac{4.8283 - 5}{0.1} \approx -1.717 \). By consulting the standard normal distribution table or using statistical software, we determine that \(P(Z > -1.717)\) is approximately 0.9564. This probability suggests a high likelihood (about 95.64%) that the ductile strength \(X\) will exceed 125.

The median is a statistical measure that indicates the middle value of a dataset or distribution. For a lognormal distribution, the median is computed simply as \(median(X) = e^{\mu}\).
In the given problem, with \(\mu = 5\), the calculation becomes \(median(X) = e^5\), which results in approximately 148.41. Interestingly, in a lognormal distribution, the median can differ significantly from the mean due to the skewness of the distribution. Here, both values end up close to each other but that's not always the case in broader contexts. Knowing the median gives us an important measure of central tendency that is less sensitive to extreme values compared to the mean.