*5. (Gaussian mixture) Let: cE (0, 1), f1, f2 denote two different Gaussian densities with param- eters µ1, oi and u2, 02, respectively, and define f(y) = cfi(y) + (1 – c)f2(y) iid Consider now...

*5. (Gaussian mixture) Let: cE (0, 1), f1, f2 denote two different Gaussian densities with param-<br>eters µ1, oi and u2, 02, respectively, and define<br>f(y) = cfi(y) + (1 – c)f2(y)<br>iid<br>Consider now Y1,..., Yn * f(y).<br>(a) Find the common mean and variance of each Y;. (Hint: find EY; and EY? by integration,<br>and use the fact that varY; = EY? – (EY:)².)<br>(b) Write an expression for the approximate sampling distribution of Y.<br>(c) For what value of c is the variance minimized?<br>

Extracted text: *5. (Gaussian mixture) Let: cE (0, 1), f1, f2 denote two different Gaussian densities with param- eters µ1, oi and u2, 02, respectively, and define f(y) = cfi(y) + (1 – c)f2(y) iid Consider now Y1,..., Yn * f(y). (a) Find the common mean and variance of each Y;. (Hint: find EY; and EY? by integration, and use the fact that varY; = EY? – (EY:)².) (b) Write an expression for the approximate sampling distribution of Y. (c) For what value of c is the variance minimized?

Jun 02, 2022

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*5. (Gaussian mixture) Let: cE (0, 1), f1, f2 denote two different Gaussian densities with param- eters µ1, oi and u2, 02, respectively, and define f(y) = cfi(y) + (1 – c)f2(y) iid Consider now...

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