Speech is sometimes idealized as having a Laplacian-amplitude pdf. That is, the amplitude is distributed according to 

f_X(x) = (a/2) exp (-a|x|)

(a) Express the variance of X, σ², in terms of a. Show your derivation; don€™t just simply copy the result given in Table 6.4.

(b) Compute the following probabilities; P(IXI > σ); P(IXI > 2σ); P(IXI > 3σ).

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Table 6.4 Probability Distributions of Some Random Variables with Means and Variances Mean Variance Probability-density or mass function Uniform: a sxsb = (x)*f 0, otherwise Gaussian: : exp |- (х - m)? /20?] = (x) XS V2ro? т Rayleigh: SR (") = exp (-r² /20²), r20 Laplacian: fx (x) = exp(-a|xl), a > 0 2/a? One-sided exponential: Sx (x) %3D а еxp (-ах) и (х) 1/a? 1/a Нуperbolic: 2h? (т — 1) h-1 m > 3, h > 0 fx (x) = 2 (|x|+ h)" * (т - 3/(т — 2) Nakagami-m: 1x 3x ...x (2m – 1) 2"T(m) Г(m + 1) 2m" fx (x) = Г(m) еxp (-тх*). х 20 Г(т) ут Central Chi-square (n = degrees of freedom)': x"/2-1 G"2n/2T (n/2) %3D 2no exp (-x/2o²) по? fx (x) = Lognormal?: •[- (Inx = m,)* /2o; еxp (гт, + а;) exp (m, + 20;) = (x) XS x/2no? X exp Binomial: P, (k) = (") p*q"-k, k = 0, 1, 2, .--, n, p+ q = 1 про пр Poisson: exp (-i), k = 0, 1, 2, .… k! P (k) = Geometric: P(k) = pqk-1, k = 1, 2, ... 9/p? 1/p