A claim count random variable N has a distribution that is a member of the (a, b,c) class. The probability mass function at k is denoted as Pk, and the probability generating function (pgf) is denoted as P(2). (a) Show that the pgf for the corresponding zero-modified distribution can be expressed as: P. M(z) = pm + 1 (P(z) – Po). Start with PM (z) = E[zNM), where NM denotes the number of claims for the corresponding zero-modified distribution. (b) Suppose N has a Poisson distribution with variance 1. Using the result from (a), show that the pgf for the corresponding zero-truncated Poisson distribution can be expressed as: PT(z) = enten

Aheat engine running backward is called a refrigerator if its purpose is to extract heat from a cold reservoir. the same engine running backward is called a heat pump if its purpose is to exhaust warm air into the hot reservoir. heat pumps are widely used for home heating. you can think of a heat pump as a refrigerator that is cooling the already cold outdoors and, with its exhaust heat qh, warming the indoors. perhaps this seems a little silly, but consider the following. electricity can be directly used to heat a home by passing an electric current through a heating coil. this is a direct, 100% conversion of work to heat. that is, 19.0 \rm kw of electric power (generated by doing work at the rate 19.0 kj/s at the power plant) produces heat energy inside the home at a rate of 19.0 kj/s. suppose that the neighbor's home has a heat pump with a coefficient of performance of 4.00, a realistic value. note: with a refrigerator, "what you get" is heat removed. but with a heat pump, "what you get" is heat delivered. so the coefficient of performance of a heat pump is k=qh/win. an average price for electricity is about 40 mj per dollar. a furnace or heat pump will run typically 200 hours per month during the winter. what does one month's heating cost in the home with a 16.0 kw electric heater? what does one month's heating cost in the home of a neighbor who uses a heat pump to provide the same amount of heating?