The article Statistical Behavior Modeling for DriverAdaptivePrecrash Systems (IEEE

Chapter 4, Problem 113

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The article Statistical Behavior Modeling for DriverAdaptivePrecrash Systems (IEEE Trans. on IntelligentTransp. Systems, 2013: 19) proposed the following mixtureof two exponential distributions for modeling thebehavior of what the authors called the criticality level ofa situation X.f(x; l1, l2, p) 5 5pl1e2l1x 1 (1 2 p)l2e2l2x x $ 00 otherwiseThis is often called the hyperexponential or mixed exponentialdistribution. This distribution is also proposedas a model for rainfall amount in Modeling MonsoonAffected Rainfall of Pakistan by Point Processes (J. ofWater Resources Planning and Mgmnt., 1992: 671688).a. Determine E(X) and V(X). Hint: For X distributedexponentially, E(X) 5 1/l and V(X) 5 1/l2; whatdoes this imply about E(X2)?b. Determine the cdf of X.c. If p 5 .5, l1 5 40, and l2 5 200 (values of the lssuggested in the cited article), calculate P(X . .01).d. For the parameter values given in (c), what is theprobability that X is within one standard deviation ofits mean value?e. The coefficient of variation of a random variable (ordistribution) is CV 5 sym. What is CV for an exponentialrv? What can you say about the value of CVwhen X has a hyperexponential distribution?f. What is CV for an Erlang distribution with parametersl and n as defined in Exercise 68? [Note: In appliedwork, the sample CV is used to decide which of thethree distributions might be appropriate.]