= Solution
For a small rectangle of side lengths $ds,dt$, <independence> makes its <probability> approximately $f_S(s)ds\,f_T(t)dt$. Thus the <joint probability density function> is the product
$$
f_{S,T}(s,t)=\frac1{\bar s\bar t}e^{-s/\bar s-t/\bar t},\qquad s,t\geq0.
$$
Adding the <probabilities> of disjoint small rectangles covering a <region of integration>, and taking their sizes to zero, gives the <double integral> of this product over that region. Without <independence>, the two <marginal distributions> alone would not determine the <joint probability density function>.
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