excans:46e9b9797e: Difference between revisions
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(Created page with "'''Solution: C''' The function <math>e(c) = E[(X-c)^2] </math> is a quadratic in <math>c</math> therefore its minimum is achieved when its derivative equals zero. Take the derivative and set it to zero: <math display = "block"> e'(c_0) =-2E[(X-c_0) = 0 \implies c_0 = E[X]. </math> Hence the minimizer equals the expected value, and therefore the answer is automatically <math>\sigma^2</math>.") |
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<math display = "block"> | <math display = "block"> | ||
e'(c_0) =-2E[(X-c_0) = 0 \implies c_0 = E[X]. | e'(c_0) =-2E[(X-c_0)] = 0 \implies c_0 = E[X]. | ||
</math> | </math> | ||
Hence the minimizer equals the expected value, and therefore the answer is automatically <math>\sigma^2</math>. | Hence the minimizer equals the expected value, and therefore the answer is automatically <math>\sigma^2</math>. | ||
Latest revision as of 12:55, 3 July 2024
Solution: C
The function [math]e(c) = E[(X-c)^2] [/math] is a quadratic in [math]c[/math] therefore its minimum is achieved when its derivative equals zero. Take the derivative and set it to zero:
[[math]]
e'(c_0) =-2E[(X-c_0)] = 0 \implies c_0 = E[X].
[[/math]]
Hence the minimizer equals the expected value, and therefore the answer is automatically [math]\sigma^2[/math].