Maximum Likelihood Estimation

luke_99

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Jun 3, 2020
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X is the result of a biased dice roll in which

[MATH] Pr(X = odd numbers) = p_o [/MATH][MATH]Pr(X = even numbers) = p_e [/MATH][MATH]3p_o + 3p_e = 1, p_o \neq p_e [/MATH]and there are n independent observations [MATH] \{X_1 , ..., X_n\} [/MATH]
How do I derive the log-likelihood function of this dataset??
I have been thinking about this problem for a while now. I would be glad if anyone could help.
 
Exactly what type of help do you need. Can we see your work so we have an idea of the method you want to use and where you need help? Please post back.
 
Thank you for the reply.
I don't fully understand the whole concept of maximum likelihood estimation, but I do know that the probability function is needed to achieve it. What I don't understand is how to get the probability function for this data set...

I found a lot of work online about deriving the maximum likelihood estimator of a Bernoulli's distribution, and I thought I could use that by changing the dice roll into a dummy [MATH]Y[/MATH] in which = 1 when [MATH]X[/MATH] is an odd number and = 0 when [MATH]X[/MATH] is an even number.
So I thought that the probability function would be
[MATH] Pr(Y_i) = 3{p_o}^{Y_i}(1-3p_o)^{1-Y_i} [/MATH]
and the likelihood function is
[MATH] L_n(p_o) = \prod_{i=1}^n3{p_o}^{Y_i}(1-3p_o)^{1-Y_i}[/MATH]
and the log-likelihood function is
[MATH] \log{L_n(p_o) }=\log{ \prod_{i=1}^n3{p_o}^{Y_i}(1-3p_o)^{1-Y_i}}[/MATH][MATH] = \sum_{i=1}^n\log{[3{p_o}^{Y_i}(1-3p_o)^{1-Y_i}]}[/MATH][MATH]= \log{3p_o}\cdot\sum_{i=1}^nY_i + \log{(1-3p_o)}\cdot\sum_{i=1}^n(1-Y_i)[/MATH]
and the maximum likelihood estimator is
[MATH] \frac{\partial \log{L_n(p_o) }}{\partial p_o}= \frac{n\mu_y}{3p_o} - \frac{n(1-\mu_y)}{1-3p_o}=0[/MATH][MATH]p_o=\frac{1}{3}\mu_y[/MATH][MATH]= \frac{1}{3n}\sum_{i=1}^nY_i[/MATH]
But I was wondering if there was a different way (possibly without changing X into Y). If I can show that the biased dice roll follows a normal distribution I could use the normal distribution function as the probability function, but I am not sure whether or not the biased dice roll does follow a normal distribution...
Anyways, here is my work. I am not sure if it makes sense or not so please give me your feedback. Thank you.
 
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