Question

1. (Memoryless property) [10]

Consider a discrete random variable X ∈ N (X ∈ N is a convention to represent that the range of X is N). It is memoryless if

Pr{X > n + m|X > m} = Pr{X > n}, ∀m,n ∈{0,1,...}.

a) Let X ∼ G(p), 0 < p ≤ 1. Show that X is memoryless. [2]

b) Show that if a discrete positive integer-valued random variable X is memoryless, it must be a geometric random variable, that is, their PMFs are identical. [3]

Now, let us consider a continuous positive random variable X, that is, its distribution function FX(t) = 0 when t < 0. It is memoryless if

Pr{X > s + t|X > s} = Pr{X > t}, ∀s,t ∈ [0,∞).

c) Let X ∼ Exp(λ), λ > 0. Show that X is memoryless. [2]

d) Show that if a continuous positive random variable X is memoryless, it must be an exponential random variable, that is, their distribution functions are identical. [3]

Interpretation: Think of X as the amount of time you wait for a bus. X, the waiting time, being memoryless simply means that no matter how long you have waited, under that condition, the probability you need to further wait for more than a certain amount of time remains the same.

Answer 1

please like this. it was so lengthy.