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10.4 Exercise 1

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Suppose X is a nonnegative, absolutely continuous random variable. Let Z=g(X)=Ce-aXZ = g \left(X\right) = C e^{- a X}, where a>0,C>0a > 0 , C > 0. Then 0<Z≤C0 < Z \leq C. Use properties of the exponential and natural log function to show that

F Z ( v ) = 1 - F X - ln ( v / C ) a for 0 < v ≤ C F_{Z} \left(v\right) = 1 - F_{X} \left(-\frac{\ln \left(v / C\right)}{a}\right) \text{for} 0 < v \leq C

Pfeiffer, Applied Probability, 10.4 Problems on Functions of Random Variables, exercise 1

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Adapted from Applied Probability © 2009 Paul E. Pfeiffer. Licensed under CC BY 3.0. Source: Applied Probability. Imported at revision d7a0420.