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**Extra info for Applied Probability and Stochastic Processes**

**Sample text**

Suppose that all distributions of the sexes for n children are equally likely. Find the probability that a family has exactly k girls. Hint: The answer is at rst as the in nite series. k+2 x2 + : : :. k ANS: 2,2 ppk+1 . 48 CHAPTER 3. 6 Show that if X 0 is a random variable such that P X 2t 10 P X t2 for all t 0; then E expjX j 1 for some 0. 7 Show that if E expX 2 = C 1 for some a 0, then for all real t. 8 Prove that function t := E maxfX; tg determines uniquely the distribution of an integrable random variable X in each of the following cases: a If X is discrete.

4. This theorem asserts that under fairly general conditions the distribution of the sum of many independent components is approximately normal. In this chapter we give another reason why normal distribution might occur. 1 on page 24. In general, the so called N m; density is given by x,m2 f x = p 1 e, 2 2 : 2 By completing the square one can check that the moment generating function M t = R 1 tZ itx Ee = ,1 e f x dx of the standard normal r. v. Z is given by 2 M t = e t2 : In multivariate case it is more convenient to use moment generating functions directly.

V. 9. Write X = X + ,RX , to decomposeRX into its non-negative, and non-positive parts, and then de ne EX = 01 P X t dt , 01 P X ,t dt. Clearly, if one of the integrals diverges, EX is not de ned. 7 Suppose X is exponential with the density f x = e,x. Let Y be X trun if X 3 cated at level 3. That is Y = X 3 if X 3 Clearly, Y is not continuous, as PrY = 3 = PrX 3 = e,3 0. On the other hand, Y is not discrete as it takes uncountable number of values; in fact all the numbers between 0 and 3 are possible.