Let {Xn}1 be a sequence of random variables. (i) Prove that if the sequence converges in probability, then the limiting random vari- able is unique P-a.s., i.e., if X, 4 X and Xn 4 Y, then P(X = Y) = 1. 4 X. (ii) Using chebyshev's inequality, show that if E [(Xn – X)²] → 0 then X,,
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![Let {Xn}1 be a sequence of random variables.
n=1
(i) Prove that if the sequence converges in probability, then the limiting random vari-
able is unique P-a.s., i.e.,
if X, 4 X and X, 4Y, then P(X = Y) = 1.
(11) Using chebyshev's inequality, show that if E [(Xn – X)²] → 0 then X,
4 x.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe8f9262-8ce4-4f34-80b8-37f063d8e88f%2Ff8a93285-9702-4961-b411-5affdbe5b464%2Friyfo_processed.png&w=3840&q=75)
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- Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.Theorem: Suppose (Xn)n≥1 is a sequence of random variables with corresponding momentgenerating functions M_Xn , and X is a random variable with moment generating functionM_X such that for some δ > 0 we have M_X (t) < ∞ for all t ∈ (−δ, δ). If lim n→∞ MXn (t) = MX (t) for all t, then lim n→∞ F_Xn (x) = F_X (x) for all x where F_X is continuous. That is, if the moment generating functions of X_n converge to the moment generating function of X, then the distribution of X_n converges to the distribution of X. Use this to show that if Sn ∼ Binomial(n, λ/n ), then the distribution of Sn converges to Poisson(λ) as n → ∞.
- 6. Consider xn+1 = (1/3)(2xn - 9/xn2). Does it converge for any nonzero initial point? If so, to what values?A simple random sample X1, …, Xn is drawn from a population, and the quantities ln X1, …, ln Xn are plotted on a normal probability plot. The points approximately follow a straight line. True or false: a) X1, …, Xn come from a population that is approximately lognormal. b) X1, …, Xn come from a population that is approximately normal. c) ln X1, …, ln Xn come from a population that is approximately lognormal. d) ln X1, …, ln Xn come from a population that is approximately normal.Modify the power series ln ( 1 + x ) = x − x 2/ 2 + x 3/ 3 − x 4 /4 + x 5/ 5 − … to find a power series for ln ( 1 + x )/ x. Then use your power series for ln( 1 + x) /x to find the power series of the antiderivative ∫ ln ( 1 + x )/ x d x.
- Let i_t denote the effective annual return achieved on an equity fund achieved between time (t -1) and time t. Annual log-returns on the fund, denoted by In(1 + i_t) , are assumed to form a series of independent and identically distributed Normal random variables with parameters u = 6% and o = 14%.An investor has a liability of £10,000 payable at time 15. Calculate the amount of money that should be invested now so that the probability that the investor will be unable to meet the liability as it falls due is only 5%. Using only formulas, no tablesA time series with a periodic component can be constructed fromxt = U1 sin(2πω0t) + U2 cos(2πω0t),where U1 and U2 are independent random variables with zero means andE(U21 ) = E(U22 ) = σ2. The constant ω0 determines the period or time ittakes the process to make one complete cycle. Show that this series is weaklystationary with autocovariance functionγ(h) = σ2 cos(2πω0hShow that the random process X(t) =cos(2π fot + θ) Where θ is an random variable uniformly distributed in the range {0, π/2, π, π/3} is a wide sense stationary process .
- Find the power series representation for the function f(x)=ln(x2+1) along with its interval of convergence.Let U1, U2, ... be i.i.d. Uni(0, 1) random variables. - log U; is distributed as exp(1). (2) Find the limit of the random sequence (U1U2 · · Un)"", as n → o. (3) Describe what happens to the sequence of random variables eVn (U¡U2 · · · Un)'/v™, (1) Show that X; as n → ∞?LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise (a) Show that the moment generating function mX(s) :=E(esX) =λ/(λ−s) for s< λ;