The Poisson distribution for a discrete variable x = 0, 1, 2, ... and real parameter λ is (a) Prove that the mean of such a distribution is (b) Prove that the variance of such a distribution is (c) The mode of a distribution is the value of x that has the maximum probability. Prove that the mode of a Poisson distribution is the greatest integer that does not exceed λ, i.e., the mode is λ. (If λ is an integer, then both λ and λ − 1 are modes.) (d) Consider two equally probable categories having Poisson distributions but with differing parameters; assume for definiteness λ1 > λ2. What is the Bayes classification decision? (e) What is the Bayes errors rate?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
The Poisson distribution for a discrete variable x = 0, 1, 2, ... and real parameter λ is
(a) Prove that the mean of such a distribution is 
(b) Prove that the variance of such a distribution is 
(c) The
(d) Consider two equally probable categories having Poisson distributions but with differing parameters; assume for definiteness λ1 > λ2. What is the Bayes classification decision?
(e) What is the Bayes errors rate?
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