 Solution

For the shifted exponential model, $F(x)=1-e^{-\lambda (x-1)}$ for $x\ge 1$. It is increasing, so the probability integral transform preserves order: $F(X_{(r)})=Y_{(r)}$. The expected uniform plotting positions are $r/(n+1)$ by part (b). Replacing the random transformed order statistic by this representative position givesThus plotting $x_{(r)}$ against $t_r=-\log (1-r/(n+1))$ should give an approximate line with intercept one and slope $\boxed{1/\lambda }$.

The replacement is a quantile-plot approximation, not an assertion that expectation commutes exactly with the nonlinear inverse distribution function. The random points need not lie exactly on the line.