Can someone please explain what a Poisson Process is and what its .pdf

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Can someone please explain what a Poisson Process is and what it\'s used for? Solution The basic form of Poisson process, often referred to simply as \"the Poisson process\", is a continuous-time counting process {N(t), t = 0} that possesses the following properties: N(0) = 0 Independent increments (the numbers of occurrences counted in disjoint intervals are independent from each other) Stationary increments (the probability distribution of the number of occurrences counted in any time interval only depends on the length of the interval) No counted occurrences are simultaneous. Consequences of this definition include: The probability distribution of N(t) is a Poisson distribution. The probability distribution of the waiting time until the next occurrence is an exponential distribution. The occurrences are distributed uniformly on any interval of time. (Note that N(t), the total number of occurrences, has a Poisson distribution over (0, t], whereas the location of an individual occurrence on t ? (a, b] is uniform.) Other types of Poisson process are described below.

Can someone please explain what a Poisson Process is and what it's used for?
Solution
The basic form of Poisson process, often referred to simply as "the Poisson
process", is a continuous-time counting process {N(t), t = 0} that possesses the following
properties: N(0) = 0 Independent increments (the numbers of occurrences counted in disjoint
intervals are independent from each other) Stationary increments (the probability distribution of
the number of occurrences counted in any time interval only depends on the length of the
interval) No counted occurrences are simultaneous. Consequences of this definition include:
The probability distribution of N(t) is a Poisson distribution. The probability distribution of the
waiting time until the next occurrence is an exponential distribution. The occurrences are
distributed uniformly on any interval of time. (Note that N(t), the total number of occurrences,
has a Poisson distribution over (0, t], whereas the location of an individual occurrence on t ? (a,
b] is uniform.) Other types of Poisson process are described below

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Cardinality Proof Problems Cardinality Proof Problems (1) Suppose f: S right arrow T, and that f is one to one. Prove that S ~ f(S). (2) Show that R~(a,b) for every a Solution f is a function from S to T f is given to be one to one i.e. each element in f has a unique image in T and also if f(a) = f(b) then a = b Hence if n is the no of elements in S, f(S) will have image for each of the n element. Hence n[f(S)] = n(S) S~f(S) ------------------------------------------------------------------------------------------------------- 2) b) Consider the function f(x) = sinx, where 0<=x<=pi/2 sin 0 =0 and sin pi/2 =1 Also sinx is one to one Hence (0,pi/2)~(0,1) c) Consider the interval (-pi/2, pi/2) as domain for the function f(x) = tan x As from -pi/2 to 0 tan is negative, we see that f is one to one. Hence f(x) = (-tan pi/2, tanpi/2) = (-infinity, infinity) Or (-pi/2, pi/2)~R.

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Capital Experts limited (CEL) specializes in taking underperforming companies to a higher level of performance.CEL\'s capital structure at December 31,2013, included 28,000 shares of $2.25 preferred stock and 225,000 shares of common stock.During 2014,CEL issued common stock and ended that year with 243,000 shares of common stockoutstanding.Average common shares outatanding during 2014 were 235,000.Income from continiing operations during 2014 was $485,000.The company discontinued a segment of the business at a loss of $89,000, and an exraordinary gain of $94,000.All amounts after income tax.Assume the number of preferred stock did not change during 2014. 1.Compute CEL\'s earnig per share .Start with the income from continuing operations 2.Analyst believe CEL can earn its current level of income for the indefinite future..Estimate the market price of a share of CEL common stock at the investment capitalization rate of 9%,11% and 13%.What estimte presumes an investment in CEL is the most risky.How can you tell Solution Since, the rate of Preferred Stock is not given. Hence,we are directly taking the Income from Conntinuing Operations for Calculation of EPS. CEL Earning Per Share = 485,000 / 235000 = 2.06.

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Carbon Factors manufactures emission detectors and employs a job-order costing system. During June, the company’s transactions and accounts included the following Raw materials purchased $265,000 Direct materials used in production 262,000 Raw materials inventory, beginning 4,200 Corporate administrative costs 21,400 Selling expenses 18,500 Sales 334,000 Total manufacturing overhead applied 39,200 Total manufacturing overhead incurred 38,100 Finished goods, beginning 17,200 Work in process inventory, beginning 13,700 Work in process inventory, ending 15,600 Direct labor cost incurred 48,000 Finished goods, ending 16,300 How much is cost of goods manufactured for June? Show all your computations. Solution Direct materials used in production $ 262,000 Direct labor cost incurred $ 48,000 Total manufacturing overhead incurred $ 38,100 Add: Work in process inventory, beginning $ 13,700 Less: Work in process inventory, ending $ 15,600 Cost ofgoods manufactured $ 346,200.

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Car Fuel Consumption Ratings Listed below are combined city-highway fuel consumption ratings (in miles/gal) for different cars measured under the old rating system and a new rating system introduced in 2008 (based on data from USA Today). The new rating were implemented in response to complaints that the old ratings were too high. Use a 0.01 significance level to test the claim the old ratings are higher than the new ratings. Old rating 16 18 27 17 33 28 33 18 24 19 18 27 22 18 20 29 19 27 20 21 New rating 15 16 24 15 29 25 29 16 22 17 16 24 20 16 18 26 17 25 18 19 Solution Old rating: n1=20, xbar1=22.7, s1=5.37 New rating: n2=20, xbar2=20.35, s2= 4.72 The test hypothesis is Ho:1=2 Ha:1>2 The test statistic is t=(xbar1-xbar2)/[s1^2/n1 + s2^2/n2] =(22.7-20.35)/sqrt(5.37^2/20 + 4.72^2/20) = 1.47 Given a=0.01, the critical value is |t(0.01, df=n1+n2-2=38)|=2.43 (check student t table) Since t=1.47 is less than 2.43, we do not reject Ho..

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Can u use the definition of convergence to prove that (5n+17)/(2n) converges.. Solution (5n+17)/(2n) I\'m going to claim that the sequence converges to 5/2. Suppose we have e > 0. We need an N such that: n > N ===> |(5n + 17) / (2n) - 5/2| < e Let\'s work backwards from |(5n + 17) / (2n) - 5/2| < e, so that each step implies the previous step: |(5n + 17) / (2n) - 5/2| < e |(5n + 17) / n - 5| < 2e |(5n + 17) / n - 5n / n| < 2e |17 / n| < 2e 17 / n < 2e ... (since n > 0) n > 17 / (2e) so it converges over n > 17 / (2e).

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1. Can someone please explain what a Poisson Process is and what it's used for? Solution The basic form of Poisson process, often referred to simply as "the Poisson process", is a continuous-time counting process {N(t), t = 0} that possesses the following properties: N(0) = 0 Independent increments (the numbers of occurrences counted in disjoint intervals are independent from each other) Stationary increments (the probability distribution of the number of occurrences counted in any time interval only depends on the length of the interval) No counted occurrences are simultaneous. Consequences of this definition include: The probability distribution of N(t) is a Poisson distribution. The probability distribution of the waiting time until the next occurrence is an exponential distribution. The occurrences are distributed uniformly on any interval of time. (Note that N(t), the total number of occurrences, has a Poisson distribution over (0, t], whereas the location of an individual occurrence on t ? (a, b] is uniform.) Other types of Poisson process are described below