from the textbook pg 153 problem 3.3
Express in words what the random variable X(subscript- N(t) +1) represents. Hint: it is the
length of which renewal interval??
Show that P{X(subscriptN(t)+1) >= x} >= F(x) (F(x) has a bar over it indicating mean)
Compute the above exactly when F(x) = 1 - e ^ -x
Solution
given that P{X(subscriptN(t)+1) >= x}>= F(x)
but by defnition of F(X) = p(X<=x) ,here X=X(subscriptN(t)+1) =XN(t)+1
so, p(XN(t)+1>=x)>= F(X)
p(XN(t)+1>=x) >= 1-e^-x (since F(x) = 1-e^-x )
1- FXN(t)+1(x) >= 1-e^-x (since F(X) = p(X<=x) = 1-p(X>=x) so we may write p(X>=x) =
1-F(X) )
- FXN(t)+1(x) >= 1-e^-x -1
FXN(t)+1(x) <= -1+e^-x+1
FXN(t)+1(x) <= e^-x
P(XN(t)+1 <= x) <= e^-x
so we have P(XN(t)+1 <= x) = e^-x and P(XN(t)+1 <= x) < e^-x
that is FXN(t)+1(x) = e^-x and FXN(t)+1(x) < e^-x