Question

4. Let X1, X2, ..., Xn be iid from the Bernoulli distribution with common probability mass function Px(x) = p*(1 – p)1-x for x = 0,1, and 0 < p < 1 14 a. (4) Find the MLE Ôule of p.

Answer 1

$p_X(x)=p^x(1-p)^{1-x}$

$X_1,X_2,....,X_n \textup{ are iid samples.}$

.:. Likelihood function L= ΠΡx (13) = ΠΡ* (1-2) - = Σκι (1-2) - Σπιτι =1 1- 1

:. Log-likelihood function = In L = In (pLi=1"(1-)"-Si=1 - (¿- ) cm p + (- - -) aucs - » i=1

$\textup{Aim is to maximise log-likelihood function in terms of }p.$

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\frac{\mathrm{d} }{\mathrm{d} p}(ln\;L)=\frac{\mathrm{d} }{\mathrm{d} p}\left[\left(\sum_{i=1}^n\!x_i\right)\!(ln \;p)+\left(n-\sum_{i=1}^n\!x_i \right )\!(ln(1-p))\right]\\ =\frac{\sum_{i=1}^n\!x_i}{p}-\frac{n-\sum_{i=1}^n\!x_i}{1-p}$

$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\frac{\mathrm{d} }{\mathrm{d} p}(ln\;L)=0\;\Rightarrow \; \frac{\sum_{i=1}^n\!x_i}{p}-\frac{n-\sum_{i=1}^n\!x_i}{1-p}=0\\ \Rightarrow \;\frac{\sum_{i=1}^n\!x_i}{p}=\frac{n-\sum_{i=1}^n\!x_i}{1-p}\\ \Rightarrow \;\frac{1-p}{p}=\frac{n-\sum_{i=1}^n\!x_i}{\sum_{i=1}^n\!x_i}\\ \Rightarrow \;\frac 1p -1=\frac{n}{\sum_{i=1}^n\!x_i}-1\\ \Rightarrow \;\frac 1p =\frac{n}{\sum_{i=1}^n\!x_i}\\ \Rightarrow \;p=\frac{\sum_{i=1}^nx_i}n=\bar x$

$\textup{MLE of }p=\hat p_{MLE}=\bar x=\textup{sample mean}$