Question

For a discrete random variable X, its hazard function is defined as hX(k) = P(X = k + 1 | X > k ) = pX(k) / 1 − FX(k)

, where FX(k) = P(X ≤ k) is the cumulative distribution function (cdf). The idea here is as follows: Say X is battery lifetime in months. Then for instance hX(32) is the conditional probability that the battery will fail in the next month, given that it has lasted 32 months so far. The notion is widely used in medicine, insurance, device reliability and so on (though more commonly for continuous random variables than discrete ones).

Show that for a geometrically distributed random variable, its hazard function is constant. We say that geometric random variables are memoryless: It doesn’t matter how long some process has been gong; the probability is the same that it will end in the next time epoch, as if it doesn’t “remember” how long it has lasted so far.

Answer 1