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Figure 2.3 The Poisson distribution describes the probability, given some mean number of random events, that x events will occur. Given a mean rate of 10 events, the probability that 10 events will occur in a particular sample is 0.125, or 12.5%â€”but on rare occasions zero events can occur.

To place these rather abstract numbers in context, an image with a sky background signal-to-noise ratio of 3 looks rough and grainy. Increasing the SNR to 10 brings a huge improvementâ€”the image still looks coarse, but it is not hopelessly bad looking. By the time the SNR has reached 30, the image appears solid, smooth, and "real." When you have gathered enough photons to reach an SNR of 100, details are clear and crisp, and nebular features appear smooth and milky.

The dependence of image quality on photon count is graphically shown in Figure 2.2, comparing synthetic images of a galaxy that have mean photon counts ranging from 0.1 to 10,000.

### 2.3.1 The Poisson and Gaussian Distributions

Although the number of photons in a sample varies, when you examine a large number of samples, a clear pattern emerges. For example, if the mean number of photons per sample is 10, the same percentage of samples will have 6 photons, the same 7, and so on. For any mean number of photons per sample, there is a well-defined probability that x photon detections will occur. For independent random events such as photon detection, the distribution of probabilities is called the Poisson distribution after the French mathematician Simeon-Denis Poisson (17811840). The following equation describes the Poisson distribution:

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