Olá helena
Seja \(X\sim Bin(3,p)\). A função verosimilhança e o seu logaritmo têm o mínimo no mesmo minimizante (x), logo sendo a função verosimilhança \(L(p|x_{1},...,x_{n})= \prod_{i=1}^{n}P(X=x_{i})\)
vamos calcular o mínimo do logaritmo da função verosimilhança. Ora
\(\frac{\partial Ln L(p|x_{1},...,x_{n})}{\partial p}= \frac{\partial Ln\left(\prod_{i=1}^{n}P(X=x_{i}) \right)}{\partial p}=\frac{\partial Ln\left(\prod_{i=1}^{n} \binom{3}{i} p^{x_{i}}(1-p)^{3-x_{i}} \right)}{\partial p}=\frac{\partial \sum_{i=1}^{n} Ln\left( \binom{3}{i} p^{x_{i}}(1-p)^{3-x_{i}} \right)}{\partial p}=\)
\(=\sum_{i=1}^{n} \frac{\partial Ln\left( \binom{3}{i} p^{x_{i}}(1-p)^{3-x_{i}} \right)}{\partial p}=\sum_{i=1}^{n} \frac{ \binom{3}{i}(x_{i} p^{x_{i}-1} (1-p)^{3-x_{i}}+p^{x_{i}}(3-x_{i}) (1-p)^{2-x_{i}}(-1))} {\binom{3}{i} p^{x_{i}}(1-p)^{3-x_{i}}}=\)
\(=\sum_{i=1}^{n} p^{x_{i}-1} (1-p)^{2-x_{i}} \left[ \frac{ x_{i} (1-p)-p(3-x_{i})} {p^{x_{i}}(1-p)^{3-x_{i}}} \right]=\sum_{i=1}^{n} \frac{ x_{i} (1-p)-p(3-x_{i})} {p(1-p)}= \frac{1}{p(1-p)}\sum_{i=1}^{n} x_{i} (1-p)-p(3-x_{i})=\)
\(=\frac{1}{p(1-p)} \sum_{i=1}^{n} x_{i}-p x_{i}-3p+p x_{i}=\frac{1}{p(1-p)} \sum_{i=1}^{n} (x_{i}-3p)=\frac{1}{p(1-p)} \sum_{i=1}^{n} (x_{i})-3np=0\)
\(\Leftrightarrow \sum_{i=1}^{n} (x_{i})-3np=0 \Leftrightarrow \sum_{i=1}^{n} x_{i}=3np \Leftrightarrow \hat{p}=\frac{\bar{x}}{3}\)
que finalmente é o Estimador de Máxima Verosimilhança de p.
Espero ter ajudado
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