Isye 2027

# Continuing for c 0 p x 2 c p c x c c c c fx

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Unformatted text preview: = 0.1, with δ = 0.02 ˆ and n = 1000. (b) If p = 0.5 and n = 1000, ﬁnd δ so P {|p−p| ≤ δ } ≈ 0.99. Equivalently, P {p ∈ [ˆ−δ, p+δ ]} ≈ 0.99. ˆ p ˆ Note that p is not random, but the conﬁdence interval [ˆ − δ, p + δ ] is random. So we want to ﬁnd p ˆ the half-width δ of the interval so we have 99% conﬁdence that the interval will contain the true value of p. (c) Repeat part (b), but for p = 0.1. (d) However, the campus network engineer doesn’t know p to begin with, so she can’t select the halfwidth δ of the conﬁdence interval as a function of p. A reasonable approach is to select δ so that, the Gaussian approximation to P {p ∈ [ˆ − δ, p + δ ]} is greater than or equal to 0.99 for any p ˆ value of p. Find such a δ for n = 1000. (e) Using the same approach as in part (d), what n is needed (not depending on p) so that the random conﬁdence interval [ˆ − 0.01, p + 0.01] contains p with probability at least 0.99 (according p ˆ to the Gaussian approximation of the binomial)? 3.7. ML PARAMETER ESTIMATION FOR CONTIN...
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