34 lines
775 B
Markdown
34 lines
775 B
Markdown
Consider the binomial distribution of the number $k$ of successes in $n$ independent trials:
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$$\sum_{i=1}^{n} X_i \sim \text{Binomial}(n, p)$$
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The log-likelihood function is:
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$$\log L(p) = \log \binom{n}{k} + k \log(p) + (n-k) \log(1-p)$$
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## Part 1: Plot Log L(p)
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Given $n=100$ and $k=10$
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![[image-1.png]]
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## Part 2: Compute the MLE
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**Given:** $\sum_{i=1}^{n} x_i = 15$ with $n = 100$ trials
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### Finding the MLE
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To find the maximum likelihood estimator, we take the derivative of the log-likelihood with respect to $p$ and set it equal to zero:
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$$\frac{d}{dp} \log L(p) = \frac{k}{p} - \frac{n-k}{1-p} = 0$$
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Solving for $p$:
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$$\frac{k}{p} = \frac{n-k}{1-p}$$
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$$k(1-p) = p(n-k)$$
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$$k - kp = pn - pk$$
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$$k = pn$$
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$$\hat{p}_{MLE} = \frac{k}{n}$$
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![[image-2.png]] |