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Each digit is a 0 or 1 with equal probability. (a) Describe a probability space { , A, P } corresponding to the contents of the register. How many points does have? (b) Express each of the following four events explicitly as a subset of the probability of these events: i. ii. iii. iv. and find The register contains 1111001100110101. The register contains exactly 4 zeros. The first 5 digits are all ones. All digits in the register are the same. 8. Chung’s disease is a heretofore unknown malady that afflicts 1 in every 100,000 Americans.

E[X · · · x f (x)dx · · · dx n] n 1 n −∞ −∞ ∞ −∞ ∞ −∞ ··· ⎫ ⎪ ⎬ ⎪ ⎭ . , E[c] = =c ∞ cf (x)dx −∞ ∞ f (x)dx −∞ 1 = c. Slightly more interesting is the fact that expectation carries over simply to a function of X. 17). Note, however, that, in general, E[g(X)] = g(E[X]). This leads to the result that the expectation operator is linear. Let Y = g(X) and Z = h(X): E[αY + βZ] = = ∞ −∞ ∞ [αg(x)f (x) + βh(x)f (x)] dx αg(x)f (x)dx + −∞ ∞ =α −∞ ∞ βh(x)f (x)dx −∞ g(x)f (x)dx + β ∞ h(x)f (x)dx −∞ = αE[Y ] + βE[Z].

24) gives us f (x, y) = = 1 (x − mX )2 (y − mY )2 exp − − 2 2πσX σY 2σX 2σY2 1 1 (x − mX )2 (y − mY )2 exp − √ exp − √ 2σX2 2σY2 σX 2π σY 2π = f (x)f (y), which implies that X and Y are independent. 8) as the product of individual density functions. Our next claim is that affine, and hence linear, combinations of Gaussians are Gaussian. 30. If X is a Gaussian random vector with mean, mX , and covariance, PX , and if Y = CX + V , where v is a Gaussian random vector with zero mean and covariance, PV , then Y is a Gaussian random vector with mean, CmX , and covariance, CPX C T + PV .