Most of our problems will start off with a statement like \"Let X1,...,Xn be a r
ID: 3352435 • Letter: M
Question
Most of our problems will start off with a statement like "Let X1,...,Xn be a random sample from the distribution." However, this distributional assumption is often just that, an assumption. Sometimes that assumption is reasonable, other times it isn't. We can sometimes use simulation, together with known properties of the assumed distribution, to check whether our distributional assumption is reasonable. To illustrate this, consider the following sample of bay anchovy larvae counts taken from the Hudson River: 158, 143, 106, 57, 97, 80, 109, 109, 350, 224, 109, 214, 84. (a) If we assume that larvae are distributed randomly and uniformly in the river, then the number collected in a fixed size net could potentially follow a Poisson distribution. Explain why this scenario matches the characteristics of a Poisson distribution. (b) To check whether or not the Poisson assumption is reasonable, we can check whether the mean and variance of the observed data are consistent with the Poisson assumption. To start with, calculate the observed mean and variance of the data. (c) If the data really do arise from a Poisson distribution, the mean and variance should be about the same (why?). In this case, they aren't. However, we wouldn't expect them to be exactly the same, even if the Poisson assumption is true. Why not? (d) The question remains, how should we expect S? to behave if the data really do arise from a Poisson distribution? To answer this question, we can carry out a simulation. We want our simulated data sets to mimic the one collected. So, we'll generate data sets of size n = 13, since that was the sample size of the original data. In addition, we'll simulate the data sets assuming they come from a Poisson distribution with l = T, since that's our best guess at the parameter. In your favorite simulation software (Excel works! So does R, SAS, Matlab, ...), generate 100 data sets each with n = 13 observations. (e) For each data set, calculate S2. (f) Construct a histogram of your 100 simulated S2 values. (g) Where does the sº from the original data set fall on this histogram? How many of the simulated S2 values are greater than the observed sº? (h) What does this lead you to believe about the Poisson assumption? Is it reasonable in this case? What is your reasoning?Explanation / Answer
Part (a)
Number of larvae in the river is as good as infinity. Thus, the size of the catch in the net can also be infinity, but the actual number may be very small. This is a typical characteristic of Poisson Distribution – infinite possibility, but low probability.
Also, catch in one net does not depend on the number already caught in the previous nets – again another typical characteristic of Poisson Distribution, ‘loss of memory’ property.
Thus. The assumption of Poisson distribution is justified. ANSWER
Part (b)
The given data is
158
143
106
109
109
350
224
109
214
57
97
80
84
The mean = 141.5385
Variance = 5880.556 ANSWER
Part (c)
Clearly mean and variance are nowhere near to each other. ANSWER
158
143
106
109
109
350
224
109
214
57
97
80
84
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