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The exponential distribution has no atoms, so all sample values are distinct almost surely, and $M_i$ is the $i$th order statistic. Set $M_0=0$ and $D_i=M_i-M_{i-1}$. Part (b) gives $D_1\sim \mathrm{Exp}(n\lambda )$.

To justify induction and the required independence, let $J$ be the label attaining the minimum. Conditional on its time and label, the remaining clocks are fresh independent rate-$\lambda$ exponential random variables, by the memoryless property. This conditional claim can be checked directly: for a specified label $j$, first ringing time $u$, and residual lifetimes $y_k>0$ for $k\ne j$, the joint probability density isThus the first minimum, its uniform winning label and the surviving residual clocks are independent. Repeat this argument after each minimum. At the $i$th step there are $n-i+1$ clocks, so the independent exponential order-statistic spacings satisfySince $Y_k/k$ has rate $k\lambda$ whenever $Y_k\sim \mathrm{Exp}(\lambda )$, summing the gaps provesIn particular the maximum is $M^{\ast }=M_n$. By variance additivity for independent random variables and the Laplace transform of an exponential distribution, its requested quantities areThe product follows from independence of the gaps, not merely from their marginal probability distributions.

The sample median is a central value of an ordered statistical sample. The usual numerical convention selects the middle order statistic for odd sample size and the arithmetic mean of the two central order statistics for even sample size:NIST uses this convention. ^[ref]. For $(0.7,0.1,0.9,0.4,0.2)$, the sample median is $0.4$; for $(0.1,0.2,0.4,0.7)$, it is $(0.2+0.4)/2=0.3$, which need not be an observed value.

More generally, every value between the two central order statistics in an even statistical sample is a median of its empirical distribution function. The generalized-inverse sample quantile at $p=1/2$ selects the lower central order statistic in that case, so it can differ from the midpoint convention. Ties are allowed in all these definitions.

The sample median measures the center using sorted positions. Changing the largest observation from $0.9$ to $9$ in the five-value example leaves the sample median at $0.4$, while the sample mean rises from $0.46$ to $2.08$. Its sampling distribution still depends on the population probability distribution: the mean $1/2$ found for five observations from a uniform distribution is a special calculation, and a sample median's expected value need not equal the population median in general.

For a concrete counterexample, three independent and identically distributed random variables with rate-one exponential distribution have sample median $X_{(2)}$ with expected value $1/3+1/2=5/6$, by independent exponential order-statistic spacings. The population median instead solves $1-e^{-m}=1/2$, giving $m=\log 2\approx 0.693147$. Thus even the middle order statistic can have mean different from the population median.