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Variance Reduction by Antithetic Variates. A simple and widely used technique for increasing the efficiency and accuracy of Monte Carlo simulations in certain situations with little additional increase in computational complexity is the method of antithetic variates. For each
Use the parameters specified in Problem 3 to compute several (say,
The difference equation (4):
Use the differential equation (4) to generate an ensemble of stock prices
where
And
The difference equation (4) is given below:
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- 1. Suppose that X1, Xn is an iid random sample from a distribution with pdf f(x; sigma) = 1/2 * (sigma) * e ^ (- |x| / sigma) - infinity0 0(a) Find the method of moments estimator (MME) of \sigma. Hint: See the example in lecture notes. (b) Find the maximum likelihood estimator (MLE) of \sigma. Hint: See the example in lecture notes. (c) Use central limit theorem to find the large - sample asymptotic distribution of the MLE in part (b). (d) Suppose 10 independent observations were taken from the given distribution: (0.5, 0.8, 2.2, 1.6, 2.0, 2.8, 1.5, 0.9, 2.5, 1.8). What is the numerical value of the MLE? What is the numerical value of the estimated standard deviation (standard error) of the MLE?arrow_forwardX1 and X2 are two discrete random variables, while the X1 random variable takes the values x1 = 1, x1 = 2 and x1 = 3, while the X2 random variable takes the values x2 = 10, x2 = 20 and x2 = 30. The combined probability mass function of the random variables X1 and X2 (pX1, X2 (x1, x2)) is given in the table below a) Find the marginal probability mass function (pX1 (X1)) of the random variable X1.b) Find the marginal probability mass function (pX2 (X2)) of the random variable X2.c) Find the expected value of the random variable X1.d) Find the expected value of the random variable X2.e) Find the variance of the random variable X1.f) Find the variance of the random variable X2.g) pX1 | X2 (x1 | x2 = 10) Find the mass function of the given conditional probability.h) pX2 | X1 (x2 | x1 = 2) Find the mass function of the given conditional probability.i) Are the random variables X1 and X2 independent? Show it. The combined probability mass function of the random variables X1 and X2 is belowarrow_forwardCONDITIONAL AND MARGINAL PROBABILITY - Statistics and probability } 1) In a plant, complex components are assembled on two different assembly lines, A and B. Line A uses older equipment than B, so it is a bit slower and less reliable. Suppose that on a given day line A assembles 8 components, of which 2 have been identified as defective (D) and 6 as non-defective (ND), while B has produced 1 defective and 9 non-defective components. . a) What is the probability that, when selecting a component at random, a defective one will come out? b) If the selected component is defective, what is the probability that it has left line A? What is the probability that it came out of line B?arrow_forward
- In the presence of enough food and lacking predators and competitors, a population of rabbitswill increase by a fixed percentage each spring. For this set of exercises, we will say that therabbit population increases by 10% each year. Thus, if the initial population is R0, then thepopulation the following spring will be R1 = (1.1)R0 rabbits. In two years, the population will beR2 = (1.1)R1 = (1.1)2R0. The number of rabbits after n years will beRn = (1.1)nR0,and clearly the rabbit population grows without bound. We will assume that the rabbit populationis measured in hundreds of rabbits (so that R = 1 represents 100 rabbits).Suppose now that we introduce a small number of cougars into the environment to keep the rabbitpopulation under control.Let’s suppose that the amount of rabbits eaten each year is proportional to the cougar population.Then the change in the rabbit population is governed by the equationRn+1 = (1.1)Rn − (0.1)Cn,where Rn and Cn represent the rabbit and cougar…arrow_forwardA time series {yt} follows an MA(2) model: Yt = 2 + Ut +0.54t-1 + 0.4ut-2. Assume that ut is a white noise series with a mean of O and a variance of 2. Please calculate Var(yt) (i.e. the variance of) 2.96 1.41 2.82 O 1.98 Please give me typed answer sirarrow_forwardLet X1, X2, ..., Xn be a sequence of independent and identically distributedrandom variables having the Exponential(λ) distribution, λ > 0,fXi(x) = λe−λx , x > 00 , otherwise(a) Show that the moment generating function mX(s) := E(e^sX) = λ/λ−s for s < λ;(b) Using (a) find the expected value E(Xi) and the variance Var(Xi).(c) Define the random variable Y = X1 + X2 +· · ·+ Xn. Find E(Y ), Var(Y ) and the moment generating function of Y .(d) Consider a random variable X having Gamma(α, λ) distribution,fX(x) = (λαxα-1/Γ(α)) e−λx , x > 00 , otherwiseShow that the moment generating function of the random variable X is mX(s) =λα 1/(λ−s)α for s < λ, where Γ(α) isΓ(α) = (integral from 0 to inifity ) xα−1e−xdx.(e) What is the probability distribution of Y given in (c)? Explain youranswer.arrow_forward
- Using 30 time series observations, the regression Y= B1 + B2 X + B3 Z + u is estimated and some results are reported as the following;Y't = 2.04 + 0.25 Xt – 0.12 Ztse (0.86) (0.08) (0.17)and the estimated first order autocorrelation coefficient (rho) P'= 0.92a) Apply Durbin-Watson d Test if there exists 1st order autocorrelation problem in the errors at 5% level (Set your hypotheses)arrow_forward1. Dickey-Fuller (ADF) test tests the null hypothesis that the series is non-stationary 2. The two most useful tools in any attempt at model identification are the sample autocorrelation function and the sample partial autocorrelation function. 3. The partial autocorrelations, of a pure moving average process decay towards zero with increasing lag length k. option: a. true b. false c. others pls scpecifyarrow_forward20 families live in a neighborhood: 4 have 1 child, 8 have 2 children, 5 have 3 children, and 3 have 4 children. Let X be the number of children in a randomly selected family in the neighbourhood. (c) Find the variance of X.(d) Find the skewness of X.(e) Find the kurtosis of X.arrow_forward
- 1)Let x be a random variable Gaussian with zero mean and variance 1. Find:a)The conditional pdf and pdf of x given x > 0;b)E [ x| x>0 ]c)Var [ x | x >0]arrow_forwardA time series {yt} follows an MA(2) model: Yt = 2 + Ut +0.54t-1 + 0.4ut-2. Assume that ut is a white noise series with a mean of O and a variance of 2. Please calculate Var(yt) (i.e. the variance of) 2.96 1.41 2.82 O 1.98arrow_forwardResistors of a certain type have resistances that are exponentially distributedwith parameter λ = 0.04. An operator connects 50 independent resistors in series, which causes total resistance in thecircuit to be the sum of individual resistances. Find the joint probability that each and everyone of them has resistance of less than 4.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage