The article "Statistical Behavior Modeling for DriverAdaptive Precrash Systems" (IEEE Trans. on Intelligent Transp. Systems, 2013: 1-9) proposed the following mixture of two exponential distributions for modeling the behavior of what the authors called "the criticality level of a situation" X.

This is often called the hyperexponential or mixed exponential distribution. This distribution is also proposed as a model for rainfall amount in "Modeling MonsoonAffected Rainfall of Pakistan by Point Processes" (J. of Water Resources Planning and Mgmnt., 1992: 671-688).
a. Determine E(X) and V(X). For X distributed exponentially, E(X) = 1/λ and V(X) = 1/λ2; what does this imply about E(X2)?
b. Determine the cdf of X.
c. If p = .5, λ1 = 40, and λ2 = 200 (values of the l's suggested in the cited article), calculate P(X > .01).
d. For the parameter values given in (c), what is the probability that X is within one standard deviation of its mean value?
e. The coefficient of variation of a random variable (or distribution) is CV = σ/μ. What is CV for an exponential rv? 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?

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