"f(k)=P(X=k)=\\binom{n}{k}\\theta^k(1-\\theta)^{n-k}"
We can look at the ratio of successive outcomes
When "r>1" then "P(X=k+1)>P(X=k)"
When "r<1" then "P(X=k+1)<P(X=k)"
Maximum of the distribution occurs when "r" switches from being greater than "1" to less than "1."
"={n-k \\over k+1}\\cdot{\\theta \\over1-\\theta}"
What value of k results in "r\\leq1?"
"\\theta(n-k)\\leq(1-\\theta)(k+1)"
"n\\theta-k\\theta\\leq k+1-k\\theta-\\theta"
"k\\geq n\\theta-(1-\\theta)"
Max probability is the smallest interger value of "k\\geq n\\theta-(1-\\theta)"
Likelihood function
Log-likelihood function
"{d({L(\\theta|n, x)}) \\over dp}=0+x\\cdot{1 \\over \\theta}+(n-x)\\cdot(-{1 \\over 1-\\theta})=0"
"x(1-\\theta)-(n-x)\\theta=0"
"x-x\\theta-n\\theta+x\\theta=0"
"x-n\\theta=0"
"\\theta={x \\over n}"
The maximum likelihood estimate for "\\theta" is just the average.
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