Derive the moment generating function of X ~ Bi(n, p) and using the MGF compute E[X] and Var[X].
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- Consider the pdf e!/3(z) 0Let c be a critical number of f(x) such that f' (c) = 0. • if f" (c) 0, then f(c) is [Select] • if f" (c) = 0, then f(c) is [Select]Find L{f (t)}using the First Shifting Theorem, Note: a, b, e, w, k and n are constants 1. L{5c * sin ot} -at Il aydc30329 2. 3.Compute the coefficient Fourier a, of the { T - r, 0Let f be a function on R with Fourier transform 1 F[f](s) = f(s) = f(x) e** dx. irs (a) Show that the Fourier transform of the function ga(x) : f(ax), where a > 0 is a constant, is Flg.|(6) = Lf (). 1 (b) Show that the Fourier transform of the function Je-, r20 f(x) = 0, x < 0 is f(s) = 1 1+ is V2n s2 +1find a control u(t) for x`(t)= [2, 1 ; 1, 1] x(t) + [1; 1] u(t), x(0)= [5; 0] so that x(2) = [0; 0] . Find the u(t) that will drive the intial state to the give final state2) For a random variable X, its kth-order moment is defined as E(X*). Given the p.d.f. fx(x) of X, its Fourier transform Fx(u) = S fx(x)e¬j2muz dx is also called moment generation function. a) Determine an expression of E(X*) in terms of Fx(u). Hint: consider the kth-order derivative of Fx (u) with respect to u and examine its relationship to the definition of E(X*). b) If X is N(0, o²), we know that Fx (u) = exp (-}(2u)°o²). Apply the above idea to determine E(X3) and E(Xª) in terms of ơ².3.) Find the DFT of f = [1 -2 1 2]!Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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