n! lim = 1, n+1/2 e n 2 n qn n in .NET Printer UPC-A in .NET n! lim = 1, n+1/2 e n 2 n qn n

n! lim = 1, n+1/2 e n 2 n qn n using visual studio .net todevelop upc barcodes with web,windows application Microsoft .NET and the formula lim 1 = e q ,. if qn q,. may be helpful. Problems 23. Let Xn b .net vs 2010 upc barcodes inomial(n, pn ) and X Poisson( ), where pn and are as in the previous problem.

Show that the probability generating function GXn (z) converges to GX (z). 24. Suppose that Xn and X are such that for every bounded continuous function g(x),.

lim E[g(Xn )] = E[g(X)].. Show that Xn c .net vs 2010 UPC-A Supplement 5 onverges in distribution to X as follows: (a) For a < b, sketch the three functions I( ,a] (t), I( ,b] (t), and 1, t < a, b t ga,b (t) := , a t b, b a 0, t > b. (b) Your sketch in part (a) shows that I( ,a] (t) ga,b (t) I( ,b] (t).

Use these inequalities to show that for any random variable Y , FY (a) E[ga,b (Y )] FY (b). (c) For x > 0, use part (b) with a = x and b = x + x to show that. lim FXn (x) FX (x + x).. (d) For x > UPCA for .NET ; 0, use part (b) with a = x x and b = x to show that FX (x x) lim FXn (x)..

(e) If x is a continuity point of FX , show that lim FXn (x) = FX (x).. 25. Show that Xn converges in distribution to zero if and only if lim E Xn 1 + . Xn = 0.. Hint: Recall P roblem 10. 26. Let f (x) be a probability density function.

Let Xn have density fn (x) = n f (nx). Determine whether or not Xn converges in probability to zero. 27.

Let Xn converge in mean of order 2 to X. Determine whether or not. n 2 lim E Xn e Xn = E X 2 e X . Other modes of convergence 28. For t 0, let Zt be a continuous-time random process. Suppose that as t , FZt (z) converges to a continuous cdf F(z).

Let u(t) be a positive function of t such that u(t) 1 as t . Show that. lim P Zt z u(t) = F(z).. Hint: Your ans wer should be simpler than the derivation in Example 14.11. 29.

Let Zt be as in the preceding problem. Show that if c(t) c > 0, then. lim P c(t) Zt z = F(z/c).. 30. Let Zt be as in Problem 28. Let s(t) 0 as t .

Show that if Xt = Zt + s(t), then FXt (x) F(x). 31. Let Nt be a Poisson process of rate .

Show that Yt :=. Nt t /t converges in d .net framework upc a istribution to an N(0, 1) random variable. Hints: By Example 14.

10, Yn converges in distribution to an N(0, 1) random variable. Next, since Nt is a nondecreasing function of t, N t Nt N t , where t denotes the greatest integer less than or equal to t, and t denotes the smallest integer greater than or equal to t. The preceding two problems and Problem 13 may be useful.

14.3: Almost-sure convergence 32. Let Xn converge almost surely to X.

(a) Show that 1 2 1 + Xn (b) Determine whether or not lim E 1 2 1 + Xn = E 1 . 1 + X2 converges almost surely to 1 . 1 + X2.

Justify your a nswer. 33. Let Xn X a.

s. and let Yn Y a.s.

If g(x, y) is a continuous function, show that g(Xn ,Yn ) g(X,Y ) a.s. 34.

Let Xn X a.s., and suppose that X = Y a.

s. Show that Xn Y a.s.

(The statement X = Y a.s. means P(X = Y ) = 0.

). Problems 35. Show that almost-sure limits are unique; i.e.

, if Xn X a.s. and Xn Y a.

s., then X = Y a.s.

(The statement X = Y a.s. means P(X = Y ) = 0.

) 36. Suppose Xn X a.s.

and Yn Y a.s. Show that if Xn Yn a.

s. for all n, then X Y a.s.

(The statement Xn Yn a.s. means P(Xn > Yn ) = 0.

) 37. If Xn converges almost surely and in mean, show that the two limits are equal almost surely. Hint: Problem 5 may be helpful.

38. In Problem 12, suppose that limn cn = c = . For each , compute the value of Y ( ) := limn cn X( ).

39. Suppose state j of a discrete-time Markov chain is transient in the sense that. n=1 (n).
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