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Exercises - Probability Theory III October 2007 1 Compute the characteristic function of a uniform distribution on the interval [a, b]. What result is obtained in the special case a = −1, b = 1? 2 Determine the distribution with characteristic function φ(t) = cos t, −∞ < t < ∞. 3 Determine the distribution with characteristic function φ(t) = t + sin t , 2t −∞ < t < ∞. 4 Determine the distribution with characteristic function 1 φ(t) = (2 cos t + 3 cos 2t + i sin 2t), 5 −∞ < t < ∞. Then compute the expected value and variance – directly and by using φ(t). 1 5 Let φ(t) be the characteristic function of a distribution symmetric around 0. Show that 1 + φ(2t) ≥ 2φ(t)2 . 6 Show that φ(t) = cos t2 , −∞ < t < ∞, cannot be the characteristic function of a random variable X with EX 2 < ∞. 7 Show that if φ(t) is a characteristic function, then exp{λ(φ(t) − 1)} is a characteristic function as well for positive λ. Hint: What is the probability generating function of a random variable with a Poisson distribution? 8 A random variable X is said to be infinitely divisible if, for each n = 1, 2, . . ., there are independent and identically distributed random variables P X1 , . . . , Xn such that ni=1 Xi has the same distribution as X. Show, by means of characteristics funcitons, that X is infinitely divisible when a) X is normally distributed with mean 0 and variance σ 2 ; b) X is Poisson distributed with mean λ. 9 Compute E(cos X) and Var(cos X) when X is normally distributed with mean 0 and variance 1. 10 Let g be a given non-negative function. a) If X is a continuous random variable with density fX , show that Z ∞ −∞ g(x)fX (x) dx < ∞ if Z ∞ −∞ 2 g 2 (x)fX (x) dx < ∞. b) Is it true that Z ∞ g(x) dx < ∞ if −∞ Z ∞ g 2 (x) dx < ∞ ? −∞ Prove or give a counterexample. 11 Let n 1X Z= Xi n i=1 be the arithmetic mean of n independent and Cauchy-distributed random variables. Show that Z is Cauchy-distributed as well. Doesn’t this violate the Law of Large Numbers? 12 Compute the density function of the continuous random variable X, if the characteristic function of X is given by φ(t) = 1 − |t| om |t| ≤ 1, 0 annars. 13 Let (X, Y ) be a bivariate random variable with finite second moments and simultaneous characteristic function φ(s, t). Suppose there exists a function g(t) such that φ(at, bt) = g(t) for all a and b such that a2 + b2 = 1. Prove that the random variables Xand Y are uncorrelated and equally distributed. 14 I the Continuity Theorem, it is required that the limit function φ of a sequence of characteristic functions is continuous at the origin in order for φ to be a characteristic function itself. Show that this requirement cannot be dropped. Hint: Consider a sequence Xn of random variables, where Xn is uniformly distributed on [−n, n]. 3 15 Suppose that X is Poisson√distributed with parameter λ. Show that asymptotically, Y = (X−EX)/ VarX is normally distributed when λ → ∞. 16 Suppose√ X has a Γ(p, a)-distribution. Show that asympotically, Y = (X − EX)/ VarX has a normal distribution when p → ∞. 17 Given a sequence Xn of random variables, where Xn ∼ Bin(n, λ/n). Show that Xn converges in distribution to a Poisson distribution. 18 Given a sequence Xn of random variables, where Xn ∼ Geom(λ/n). Show that Xn /n converges in distribution to an exponential distribution. 19 Let X and Y be two independent symmetric random variables such that X + Y and X − Y are independent as well. (Symmetry of X means that X and −X have the same distribution.) Show, e.g. by means of characteristic functions, that X and Y have the same distribution. 20 Let Xi be a sequence of independent random variables, all having a uniform distribution on [0, θ]. Define Yn = min{X1 , X2 , . . . , Xn }, n ≥ 1. Show that nYn converges in distribution and write down the limit distribution. 21 Suppose that Xn is a sequence of random variables converging to X in distribution, and that an is a sequence of positive numbers converging to 0 as n → ∞. Show that an Xn converges to 0 in probability as n → ∞. 4 22 Suppose Xn is a sequence of random variables converging almost surely to X, and that cn is a sequence of numbers converging to 0 as n → ∞. Show that Xn + cn converges almost surely to X as n → ∞. 23 Let Xn be a sequence of independent random variables, defined on the same probability space, with P(Xn = n) = pn and P(Xn = 0) = 1 − pn . Give conditions on {pn } that imply convergence of Xn to 0 a) almost surely (you only need to give a sufficient condition); b) in rth moment; c) in probability; d) in distribution. 24 Let Xi be a sequence of independent random variables with expected value 0 and VarXi ≤ M for all i. Show that the sequence n 1X Yn = Xi n i=1 of arithmetic means converges to 0 in probability as n → ∞. 25 Let Xi be a sequence of independent random variables, uniformly distributed on [0, 1]. Show that the sequence Yn = (X1 X2 X3 · · · Xn )1/n of geometric means converges in probability to a certain constant as n → ∞. Which constant? 5 26 Let Xi be a sequence of independent random variable, all uniformly P distributed on [0, θ]. Show that max{X1 , X2 , . . . , Xn } → θ. 27 Let Xi be a sequence of random variables with expected value 0 and the same finite variance. Suppose, in addition, that Cov(Xi , Xj ) ≤ 0 if i 6= j. Show that the sequence n 1X Yn = Xi n i=1 converges to 0 in probability. 28 Prove or give a counterexample: n.s. n.s. n.s. a) Xn −→ X and Yn −→ Y imply Xn + Yn −→ X + Y . r r r P P P D D D b) Xn → X and Yn → Y imply Xn + Yn → X + Y . c) Xn → X and Yn → Y imply Xn + Yn → X + Y . d) Xn → X and Yn → Y imply Xn + Yn → X + Y . 29 Prove or give a counterexample: n.s. n.s. n.s. a) Xn −→ X and Yn −→ Y imply Xn Yn −→ XY . r r r P P P D D D b) Xn → X and Yn → Y imply Xn Yn → XY . c) Xn → X and Yn → Y imply Xn Yn → XY . d) Xn → X and Yn → Y imply Xn Yn → XY . 6 30 Let A1 , A2 , . . .be a sequence of independent events such that P (Ai+1 ) ≥ i P (Ai ) i+1 for i = 1, 2, . . .. Show that P (Ai i.o.) is either 0 or 1. 31 Let X and Y be random variables defined on the same probability space such that X is N (0, 1)-distributed and Y is Cauchy-distributed. According 1 n.s. P D to which convergence criteria →, →, → and → do the sequences Zn = an X + bn Y converge for different a and b such that −1 ≤ a, b ≤ 1? P P 32 Let g : R → R be continuous. Show that g(Xn ) → g(X) if Xn → X. 33 Let g : R → R be continuous and bounded. Show that Eg(Xn ) → D Eg(X) if Xn → X. Hint: Use Skorokhod’s Representation Theorem. 34 Let Yj be independent and identically distributed random variables, each one uinformly distributed on {0, 1, . . . , 9}. Show, e.g. by means of characteristic functions, that the sequence Xn = n X Yj 10−j j=1 converges in distribution to a uniform distribution on [0, 1]. Then show that n.s. Xn −→ X for some X that is uniformly distributed on [0, 1]. 7 35 Suppose ∞ X E |Xn − X|r < ∞ for some r > 0. n=1 n.s. Show that Xn −→ X. 36 Prove the following version of the Strong Law of Large Numbers: If Xi is a sequence of independent and identically distributed random variables with expected value 0 and finite fourth moment, EX14 < ∞, then Yn = n 1X n.s. Xi −→ 0. n i=1 37 If X has mean 0 och variance σ 2 , show that P(X ≥ t) ≤ σ2 , σ 2 + t2 for t ≥ 0. Hint: Introduce Y = X + c and then use Chebyshev’s Inequality. 8