Question: Let ????1, ????2 be two constants with 0 ????1, ????2 1 and define the probability function f (x) by the formula f (x)

Let ????1, ????2 be two constants with 0 ≤ ????1, ????2 ≤ 1 and define the probability function f (x) by the formula f (x) = ????1f1(x) + ????2f2(x), x = 0, 1, 2,…

Find the condition that must be satisfied by the ????i’s so that f is a valid probability function of a random variable, which takes values on the nonnegative integers.

Then, find the mean and variance of the resulting distribution.

Application: In an automobile insurance portfolio, the insurance company estimates that the number of claims made per year by a male driver has a Poisson distribution with parameter ????1, while the number of claims made per year by a female driver has a Poisson distribution with parameter ????2. If it is known that 60% of the insured drivers are males, find the probability function and the expected value of the number of accidents per driver (either male or female) in this portfolio.

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