A daily count takes the values 0, 1 or 2 and has mean 10/7. The maximum-entropy law with that mean has the shape p(k) ∝ r^k for some r > 0. Find P(0), given that r is a whole number. Give a fraction.

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A daily count takes the values $0$ , $1$ or $2$ and has mean $\frac{10}{7}$ . The maximum-entropy law with that mean has the shape $p (k) \propto r^{k}$ for some $r > 0$ . Find $P (0)$ , given that $r$ is a whole number. Give a fraction.

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Hint

Write the mean as a function of $r$ and try small values.

Solution

The mean is $\frac{r + 2 r ^{2}}{1 + r + r ^{2}}$ . With $r = 2$ it is $\frac{2 + 8}{1 + 2 + 4} = \frac{10}{7}$ .
So the weights are $1$ , $2$ and $4$ , and $P (0) = \frac{1}{1 + 2 + 4} = \frac{1}{7}$ .

Answer: $\frac{1}{7}$

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