Question (Exponential Family [5 marks]). Consider that \( X_{1}, \cdots, X_{n} \) are independent and identically distributed random variables with the probability density function \[ f_{X}(x)=\frac{\Gamma(2 \alpha)}{\Gamma(\alpha)^{2}}(x(1-x))^{\alpha-1} . \]
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The given probability density function (pdf) is: \[ f_X(x) = \frac{\Gamma(2\alpha)}{\Gamma(\alpha)^2} (x(1-x))^{\alpha-1} \] This pdf is defined for \( x \) in the interval [0, 1]. The form of the function \( (x(1-x))^{\alpha-1} \) suggests that it might be Show more…
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