The kurtosis of the leptokurtic Poisson distribution with \(\lambda = 0.5\) is \(K = 19\).
The kurtosis (\(K\)) of a probability distribution measures the tail heaviness or sharpness of the distribution. For a Poisson distribution, the kurtosis is given by the formula:
\[ K = \frac{y_4 - 3(y_2)^2 + 3 }{(y_2)^2} \]
where \(y_k\) is the \(k\)-th central moment. For a Poisson distribution with parameter \(\lambda\), the central moments are \(y_2 = \lambda\) and \(y_4 = 3\lambda + 1\).
Given that \(\lambda = 0.5\), substitute these values into the formula:
\[ K = \frac{(3(0.5) + 1) - 3(0.5)^2 + 3}{(0.5)^2} \]
\[ K = \frac{2.5 - 3(0.25) + 3}{0.25} \]
\[ K = \frac{2.5 - 0.75 + 3}{0.25} \]
\[ K = \frac{4.75}{0.25} \]
\[ K = 19 \]
Therefore, the kurtosis of the leptokurtic Poisson distribution with \(\lambda = 0.5\) is \(K = 19\).