FM5002-HW12-5.2.12

Fm5002hw125212nb f3 t e f 4 t i a3 e ia t a4 e i b3

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Unformatted text preview: ary PCRV such that Pr[X = a] = p, Pr[X = b] = q, an d Pr[X = c] = r, where p + q + r = 1. Let f (t) be the Fourier 4 tran sform of the distribution of X. a. Compute E X4 . b. Compute E X5 . c. Compute f (0). d. Compute f 4 (0). e. Compute f 5 (0). a. E X4 p a4 q b4 r c4 . b.E X5 pa5 q b5 r c 5. c. The generatin g function for the distribution of Xis p z a q z b r z c, so that the Fourier tran sform is f t p e i a t q e i b t r e i c t an d f 0 p q r 1. d . Atedious computation or observation of the periodicity of the derivatives gives f 4 0 e. Similarly, f 5 0 i a5p b5 q a4 p b4 q c5 r . 0059− . Let X be a PCRV whose distribution satisfies : Pr[X = −2] = 0.4, Pr[X = 0] = 0.4, an d Pr[X = 4] = 0.2. 5 a. Fin d the gen erating fun ction of the distribution of X. b. Fin d the Fourier tran sform of the distribution of X. c. Let X1, X2, ... be an i.i.d. sequen ce of PCRVs, all with the same distribution as X. Fin d the Fourier tran sform of the distribution of X1 X2...
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