4. (18 pts) Suppose that a person's daily income, X₁, on day i for i € {1,..., n}, is random. Assume that on day i, the

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4 18 Pts Suppose That A Person S Daily Income X On Day I For I 1 N Is Random Assume That On Day I The 1 (153.2 KiB) Viewed 54 times

4. (18 pts) Suppose that a person's daily income, X₁, on day i for i € {1,..., n}, is random. Assume that on day i, the person spends of his/her daily income of day i - 1. Let Y₁ be the person's net income on day i, Y₁ = X₁ − ³ Xi-1; Assume that Xo = 0. (a) Assume that the person's daily income, X₁, are i.i.d random variables with mean 500 and variance 10000. Find the mean, autocorrelation, and autocovariance functions of Y₁. (b) Assume that the person's daily income, X₁, are taken from a Gaussian wide-sense stationary random process with mean 500 and autocovariance function given by Cx(j, k) = 10000e-³|j-kl. Find the mean, variance, autocorrelation and autocovariance functions of Y₁. (c) Find an expression for the joint probability density function of Y₂ and Y5 in (b). Hint: Use linearity of the expectation.