2.6 Apply the inverse-transform method to generate random variables from a Laplace distribution (i.e., a shifted two-sided exponential distribution) with pdf -∞ < x <∞ (A > 0). f(x) = = 1 2 -X|x-0| e
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- A discrete random variable X has probability mass function X 0 1 2 3 4 P(x) 0.1 0.2 0.2 0.2 0.3 Use the inverse transform method to generate a random sample of size 1000 from the distribution of X. Construct a relative frequency table and compare with the theoretical values.A random variable X has a pdf SC(1-x*) -1Let X be a random variable with pdf f(x) = 0.2 €-0.2x 0 if x > 0, otherwise. (a) Find the moment generating function of X, remembering to state the range for which it holds. (b) Hence find the expectation and variance of X.(11) Find the moment-generating function (MGF) of an exponential random variable with param- eter A.x is a gaussian random variable with a PDF as described above, where μ is the mean, σ is the standard deviation , and Fx(X) refers to the cumulative distribution function CDF. It is known that Fx(-0.7) = 0.500 and Fx(1.3)=0.841, what value of Xo do we find the probability Fx(Xo) = P(X<Xo) = 0.023 ?der a random sample X1, X2, ..., X, having the pdf, f(r; 0) 1A 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 1.988. Let the random variable X have the pdf 2 x2 fx (x) = exp %3D - V2n 2 Find the mean and the variance of X. Hint: Compute E (X) directly and E (X²) by comparing the integral with the integral representing the variance of a random variable that is N(0,1). i DCO 04 < (X - 5)2 < 38.4).4. Using the inverse CDF method, find formulae for generating random variables having the following PDFs: (a) f(x) = 2 cos x 3 sin³ x " (b) f(x) = 8x (x + 1)³ 0≤x≤1. 3"Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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