Normal approximation to the Poisson distribution method of estimating a rate l

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A Poisson(a) has a mean and standard deviation given by:

From Central Limit Theorem, as a gets large:

Equation 2 for the Poisson distribution method can then be rewritten:

                                   (1)

Figure 1: Example of Equation 1 estimate of l where a = 40, t = 4

Figure 2: Example of Equation 3 estimate of l where a = 2, t = 4

Equation 1 works nicely in Figure 1 for large a (40): a is the measure of the amount of data, whereas t is just a scaling factor. Figure 2 shows the normal distribution approximation is less useful for smaller a (2): now Equation 1 is completely inaccurate, assigning considerable confidence to negative values, and fails to reflect the asymmetric nature of the uncertainty distribution.