# 程序代写代做代考 algorithm PDF document created by PDFfiller

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Problem 1

Figure 1

Set up:

Consider the data set NormalMix.csv and its histogram displayed in Figure 2. The above

histogram shows a clear bimodal shape in the distribution of X. One way to model a

distribution of this type is to use a mixture of two probability distributions. Here we assume

that our data set NormalMix.csv is a random variable governed by the probability density

f(x), defined by

f(x) = f(x;µ1, σ1, µ2, σ2, δ)

= δf1(x;µ1, σ1) + (1− δ)f2(x;µ2, σ2)

= δ
1

2πσ2
1

exp−
1

2σ2
1

(x− µ1)
2 + (1− δ)

1

2πσ2
2

exp−
1

2σ2
2

(x− µ2)
2,

where −∞ < x < ∞ and the parameter space is defined by −∞ < µ1, µ2 < ∞, σ1, σ2 > 0,

and 0 ≤ δ ≤ 1. The mixture parameter δ governs how much mass gets placed on the first

distribution f(x;µ1, σ1) and the complement of δ governs how much mass gets placed on

the other distribution f2(x;µ2, σ2).

2

In our setting, we have n = 10, 000 sampled observations but we do not know how many

males and females were sampled. Assume that the distribution of males is governed by

f1(x;µ1, σ1) =
1

2πσ2
1

exp−
1

2σ2
1

(x− µ1)