Let \(X\) be a random variable with range \(\{0,1,2,3,\dots\}\) and distributed geometrical with probability \(p\).

1. Show that for every \(n \geq 0\) and \(j\geq 0\),
   \[ \mathbf{P}(X-n=j \mid  X\geq n) = \mathbf{P}(X=j)\]
2. If \(X \) is the time to the failure of a machine, then \(\mathbf{P}( X\geq n) \) is the event that the machine has not failed by time \(n\).  Why is the above property called Memorylessness ?
3. Show that the geometric distribution is the only random variable with range equal to \(\{0,1,2,3,\dots\}\) with this property.