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Assign_2_2010_Solutions

# Assign_2_2010_Solutions - STAT 230 Assignment 2 Solutions 1...

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STAT 230 - Assignment 2 Solutions 1. Suppose X is a random variable with probability function f x kx 0.3 x 1 , x 1,2, a Given that f x is a probability function, find k . Since f x is a probability function k x 1 x x 1 1. Since 1 t t 2 t 3 x 0 t x 1 1 t , | t | 1 by the Binomial series then by the theorem on the differentiation of power series d dt x 0 t x d dt 1 1 t , | t | 1 or x 1 xt x 1  1 1 t 2 , | t | 1. Let t 0.3 to obtain 1 k x 1 x x 1 1 1 2 1 0.7 2 1 0.49 and thus k 0.49. b Find E X . Multiply both sides of x 1 xt x 1 1 1 t 2 , | t | 1. by t to obtain x 1 xt x t 1 t 2 , | t | 1. Differentiate both sides with respect to t and use the theorem on the differentiation of power series to obtain x 1 x 2 t x 1 1 t 2 t 2  1 t 1 t 4 1 t 1 t 3 , | t | 1. Therefore E X x 1 x x x 1 x 1 x 2 x 1 1 1 3 13 7 .

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2. Suppose it is assumed that the number of M7 earthquakes in a year follows a Poisson process with intensity, , measured in earthquakes per year. a What three assumptions does this make about the occurrences of earthquakes? How likely is each of the assumptions? Independence: The number of M7 earthquakes during one time interval does not affect the number of M7 earthquakes during any non-overlapping time interval. Individuality: There are never two M7 earthquakes at exactly the same time. Homogeneity: For short time intervals, the probability that there is M7 earthquake during a given interval is proportional to the length of that interval and the rate of occurrence is homogeneous over time. The crust of the Earth is made up of about a dozen pieces called tectonic plates. These plates “float” on the melted core of the Earth. As the plates shift, they grind up against each other. This builds up pressure between the plates until the plates eventually slip and energy is released in what is called an earthquake. Following the main shock there may be further movements, called aftershocks, which occur as the plates “settle down” in their new positions. From this description of what causes earthquakes it seems plausible that the assumptions of independence and individuality hold since the pressure building up at one place where the two plate edges are grinding is independent of the pressure building up at other places where two plate edges are grinding together. It is also believed that earthquakes do not cause other earthquakes but that earthquakes are followed by a series of aftershocks. Homogenity of the rate of M7 earthquakes over the Earth’s crust may not hold since plates are moving at different rates.
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Assign_2_2010_Solutions - STAT 230 Assignment 2 Solutions 1...

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