MLE - Bernoulli Distribution

Let {𝑋₁, ..., 𝑋_𝑛} be samples taken from a Bernoulli(𝑝) Distribution

How to estimate parameter 𝑝 using MLE method?

the log-likelihood function 𝓛(𝜃) of 𝑛 Bernoulli Distribution is as follows

• 𝓛(𝑝) = 𝑘·𝑙𝑛(𝑝) + (𝑛-𝑘)·𝑙𝑛(1-𝑝) # click here for step-by-step computation
  

now differentiate with respect to 𝑝

• 𝓛(𝑝) = 𝑘·𝑙𝑛(𝑝) + (𝑛-𝑘)·𝑙𝑛(1-𝑝)
• 𝓛'(𝑝) = 𝑘/𝑝 - (𝑛-𝑘)/(1-𝑝)

equate to 0 and solve for 𝑝

• 𝓛'(𝑝) = 𝑘/𝑝 - (𝑛-𝑘)/(1-𝑝)
• 0 = 𝑘/𝑝 - (𝑛-𝑘)/(1-𝑝)
• (𝑛-𝑘)/(1-𝑝) = 𝑘/𝑝
• (𝑛-𝑘)𝑝 = 𝑘(1-𝑝)
• 𝑛𝑝 - 𝑘𝑝 = 𝑘 - 𝑘𝑝
• 𝑛𝑝 - 𝑘𝑝 + 𝑘𝑝 = 𝑘
• 𝑛𝑝 = 𝑘
• 𝑝_𝑀𝐿𝐸 = 𝑘 / 𝑛