Operation \[F(r)\] Description
\[t^{n}f(t)\] \[{( - 1)}^{n}\frac{d^{n}F(r)}{dr^{n}}\] Used to find moments
\[e^{at}f(t)\] \[F(r - a)\] Discounting produces a shift in the transform distribution
\[f'(t)\] \[rF(r) - f(0)\] Derivative: Used to solve differential equations
\[f*g = \int_{0}^{t}{f(x)g(t - x)}\] \[F(r)G(r)\] Convolution: Sum of independent random variables
\[f*\delta(t - t_{0})\] \[e^{- rt_{0}}F(r)\] Shifted output, or delay
\[f*1\] \[\frac{F(r)}{r}\] Integral Operator