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A random variable takes the values $1$ , $2$ and $3$ with probabilities $θ$ , $2 θ$ and $1 - 3 θ$ , where $0 < θ < \frac{1}{3}$ . In $20$ independent observations, the value $1$ appeared $6$ times, the value $2$ appeared $4$ times and the value $3$ appeared $10$ times. What is the maximum likelihood estimate of $θ$ ?

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Hint

Collect the powers of θ and of $1$ − 3θ in the likelihood before differentiating.

Solution

The likelihood is proportional to $θ^{6} (2 θ)^{4} (1 - 3 θ)^{10}$ , so the log-likelihood is $10 ln θ + 10 ln (1 - 3 θ)$ plus a constant.
Its derivative is $10/ θ - \frac{30}{1 - 3 θ}$ .
Setting it to zero gives $10 (1 - 3 θ) = 30 θ$ , so $10 = 60 θ$ .
The estimate is $θ = \frac{1}{6}$ .

Answer: $\frac{1}{6}$

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