EN 257 : Applied Stochastic Processes Problem Set 5 Douglas
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چکیده
The members of a sequence of jointly independent random variables X[n] have probability density functions of the following form. f X (x; n) = 1 − 1 n 1 √ 2πσ exp − 1 2σ 2 x − n − 1 n σ 2 + 1 n σ exp(−σx)u(x) Determine whether or not the random sequence X[n] converges in (a) the mean-square sense, (b) probability, (c) distribution. Recall, from Definition 6.7-5 on page 379 in [4], that a random sequence X[n] converges in the mean-square sense to the random variable X if lim n→∞ E{|X[n] − X| 2 } = 0. Following the derivation on page 420-421 in [2], we note the Cauchy criterion requires that the following condition must hold in order for mean-square convergence. lim n→∞ E{|X[n] − X| 2 } = 0 ⇐⇒ lim n→∞, m→∞ E{|X[n] − X[m]| 2 } = 0 For the real-valued random sequence X[n], we have the following result. E{|X[n] − X[m]| 2 } = E{(X[n] − X[m]) 2 } = E{X[n] 2 } − 2E{X[n]X[m]} + E{X[m] 2 } = E{X[n] 2 } − 2E{X[n]}E{X[m]} + E{X[m] 2 }, for n = m (1) Note that in the previous expression we have substituted E{X[n]X[m]} = E{X[n]}E{X[m]}, since {X[n]} are jointly independent and the expression will be nonzero only for the case n = m. Substituting the integral expressions for the expectations, we find
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