# Understanding the Poisson Memorylessness Proof

I was hoping someone can explain to me step by step the proof of Poisson memorylessness property. First i understand that ,

1. Memoryless: P(X>s + t|X>t) = P(X>s).


From that point the rule of conditional probability is used which will then simplify to :

2.P(X>s + t, X > t)/P(X>t)


But from this point, i don't understand the mathematics involved in these subsequent phases. Please explain as you would to a novice how each step is simplified, my background in mathematics is average.

3.P(X>s + t)/P(X>t)
4.e−λ(s+t)/e−λt
5.e−λs
6.P(X>s)

• Step 3. If $X>t$ and it is also true that $X > s+t$, then, for $s > 0$, is it not correct that the first statement is redundant; since $X > s+t$, we already know that $X > t$ without needing to be told this separately. Jun 17 '14 at 11:56

3. $[X\gt s+t,X\gt t]=[X\gt s+t]$ since $s\geqslant0$.

4. Definition of exponential CDF is $P(X\gt x)=\mathrm e^{-\lambda x}$ for every $x\geqslant0$.

5. Obvious.

6. Definition of exponential CDF.

• Thanks i now understand. The CDF was the missing piece. Please if this doesn't irritate you, i don't get the most simple obvious step 5. Sorry :( Jun 17 '14 at 12:40
• ?? What do you know about the exponential function?
– Did
Jun 17 '14 at 12:42
• i feel stupid right now. I now know why :D If we are dividing exponents of the same base we can just subtract the powers Jun 17 '14 at 12:53