Discrete-time stochastic processes

Exercise 117 a show that for uncorrelated rvs the

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: integral as a sum where X is a non-negative integer random variable. b) Generalize the above integral for the case of an arbitrary (rather than non-negative) rv Y with distribution function FY (y ); use a graphical argument. c) Find E [|Y |] by the same type of argument. d) For what value of α is E [|Y − α|] minimized? Use a graphical argument again. Exercise 1.8. a) Let Y be a nonnegative rv and y > 0 be some fixed number. Let A be the event that Y ≥ y . Show that y IA ≤ Y (i.e., that this inequality is satisfied for every ω ∈ ≠). b) Use your result in part a) to prove the Markov inequality. Exercise 1.9. Use the definition of a limit in the proof of Theorem 1.2 to show that the sequences in parts a and b satisfy limn→1 an = 0 but the sequence in part c does not have a limit. a) an = 1 ln(ln n) b) an = n10 exp(−n) c) an = 1 for n = 10k for each positive integer k and an = 0 otherwise. d) Show that the definition can be changed (with no change in meaning) by replacing δ with either 1/m or 2−m for every positive integer m. 52 CHAPTER 1. INTRODUCTION AND REVIEW OF PROBABILITY Exercise 1.10. Let X be a rv with distribution function FX (x). Find the distribution function of the following rv’s. a) The maximum of n IID rv’s with distribution function FX (x). b) The minimum of n IID rv’s with distribution FX (x). c) The difference of the rv’s defined in a) and b); assume X has a density fX (x). Exercise 1.11. a) Let X1 , X2 , . . . , Xn be rv’s with expected values X 1 , . . . , X n . Prove that E [X1 + · · · + Xn ] = X 1 + · · · + X n . Do not assume that the rv’s are independent. b) Now assume that X1 , . . . , Xn are statistically independent and show that the expected value of the product is equal to the product of the expected values. c) Again assuming that X1 , . . . , Xn are statistically independent, show that the variance of the sum is equal to the sum of the variances. Exercise 1.12. Let X1 , X2 , . . . , Xn , . . . be a sequence of IID continuous rv’s with the common probability density function fX (x); note that Pr {X =α} = 0 for all α and that Pr {Xi =Xj } = 0 for all i 6= j . For n ≥ 2, define Xn as a record-to-date of the sequence if Xn > Xi for all i < n. a) Find the probability that X2 is a record-to-date. Use symmetry to obtain a numerical answer without computation. A one or two line explanation should be adequate). b) Find the probability that Xn is a record-to-date, as a function of n ≥ 1. Again use symmetry. c) Find a simple expression for the expected number of records-to-date that occur over the first m trials for any given integer m. Hint: Use indicator functions. Show that this expected number is infinite in the limit m → 1. Exercise 1.13. (Continuation of Exercise 1.12) a) Let N1 be the index of the first record-to-date in the sequence. Find Pr {N1 > n} for each n ≥ 2. Hint: There is a far simpler way to do this than working from part b in Exercise 1.12. b) Show that N1 is a rv. c) Show that E [N1 ] = 1. d) Let N2 be the index of the second record-to-date in the sequence. Show that N2 is a rv. Hint: You need not find the distribution function of N2 here. e) Contrast your result in part c to the result from part c of Exercise 1.12 saying that the expected number of records-to-date is infinite over an an infinite number of trials. Note: this should be a shock to your intuition — there is an infinite expected wait for the first of an infinite sequence of occurrences. 1.8. EXERCISES 53 Exercise 1.14. (Another direction from Exercise 1.12) a) For any given n ≥ 2, find the probability that Nn and Xn+1 are both records-to-date. Hint: The idea in part b of 1.12 is helpful here, but the result is not. b) Is the event that Xn is a record-to-date statistically independent of the event that Xn+1 is a record-to-date? c) Find the expected number of adjacent pairs of records-to-date over the sequence X1 , X2 , . . . . 1 1 Hint: A helpful fact here is that n(n1 = n − n+1 . +1) Exercise 1.15. a) Assume that X is a discrete rv taking on values a1 , a2 , . . . , and let Y = g (X ). Let bi = g (ai ), i≥1 be the ith value taken on by Y . Show that E [Y ] = P P i bi pY (bi ) = i g (ai )pX (ai ). b) Let X be a continuous rv with density fXR x) and let g be differentiable and monotonic ( R increasing. Show that E [Y ] = y fY (y )dy = g (x)fX (x)dx. Exercise 1.16. a) Consider a positive, integer-valued rv whose distribution function is given at integer values by FY (y ) = 1 − 2 (y + 1)(y + 2) for integer y ≥ 0 Use (1.24) to show that E [Y ] = 2. Hint: Note the PMF given in (1.21). b) Find the PMF of Y and use it to check the value of E [Y ]. c) Let X be another positive, integer-valued rv. Assume its conditional PMF is given by pX |Y (x|y ) = 1 y for 1 ≤ x ≤ y Find E [X | Y = y ] and show that E [X ] = 3/2. Explore finding pX (x) until you are convinced that using the conditional expectation to calculate E [X ] is considerably easier than using pX (x). d) Let Z be another integer-valued rv with the conditional PMF pZ |Y (z |y ) = 1 y2 for 1 ≤ z ≤ y 2 Find E [Z | Y = y ] for each integer y ≥ 1 and find E [Z ]. Exercise 1.17...
View Full Document

Ask a homework question - tutors are online