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From your knowledge of the relationships among the various cost func.pdf

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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.

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- 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

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