Question

Let X be an exponential r.v. (a) Find and plot Fx (2|X >t). How does Fx(2|X > t) differ from Fx()? (b) Find and plot fx(x|X >t). (c) Show that P[X >t+x|X > t] = P[X > x). Explain why this is called the memoryless property.

Answer 1

Answer : Given that e co 220 X is an Cxponential random Variable. a) Find and plot Fx («IX>+) How does Fx (/x ) diffen Fula). The Cdf Fx (x) of an expotential random Variable is. Fx (0) Eileri x 20 Fx (xlx>+) - P[{X<x31 {X3+3] P(x >+] :52[) - Fu+) - I-Fr (7) ust xz7 so (l-edu-li-e-it) X5+ -(1-e-It) xst

x2t Leyt iety x >7 ext Thus, Fx (2/X>t) is the delayed version..of Fr (2) The plot of f (xlx>t) is geven as: Fx (2) > : b) To Find: fox (alX 3+) The paf fx (x), of an Potential random Variable is:

fx (a)= so x ..O leta azo NOW, fore Gelx> +) fx (x) nl χ2) |- Fx (+) = le-in 1-(1-0) Xt De A (at) azt The plot of for (xlxst) is geven as c) To Show : P[X>++/X3+)- P[x>x] P[x>++ x (x > +- P [x>++ x3 n{X>0}] P(x>7

= 1-Fx (tta) I-Fx C+) -1-(1-0-2 (4x2)) 1-11-02) ce(+) etz =e- Axi . = = P(xxx] This is called the memoryless prioperty Since the result above explains that, the additional waiting time doy not depend on the time already spent waiting