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Definition: Gamma distribution is a distribution that arises naturally in processes for which the waiting times between events are relevant. It can be thought of as a waiting time between Poisson distributed events. Probability density function: The waiting time until the hth Poisson event with a rate of change Î» is.
The. \(\Gamma \) is defined as follows \[ \Gamma(k) = \int_0^\infty x^{k-1} e^{-x} \, dx, \quad k \in (0, \infty) \] The function is well defined, that is, the integral converges for any \(k \gt 0\).On the other hand, the integral diverges to \(\infty \) for \(k \le 0 \).uppose that the lifetime of a device (in hours) has the gamma distribution with shape parameter \(k = 4\) and scale parameter \(b = 100\). 1  Find the probability that the device will last more than 300 hours. 2  Find the mean and standard deviation of the lifetime.