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Q54Section C (Section C)54 of 60

MLE of gamma population variance

Let x1=2.1,x2=4.2,x3=5.8x_1=2.1, x_2=4.2, x_3=5.8 and x4=3.9x_4=3.9 be the observed values of a random sample X1,X2,X3X_1, X_2, X_3 and X4X_4 from a population having a probability density function f(x;θ)={xθ2e−xθ,if x>0, 0,otherwise,f(x; \theta)=\begin{cases} \frac{x}{\theta^2} e^{-\frac{x}{\theta}}, & \text{if }x > 0, \ 0, & \text{otherwise,} \end{cases} where θ∈(0,∞)\theta\in(0, \infty) is an unknown parameter. Then, the maximum likelihood estimate of Var(X1)\mathrm{Var}(X_1) equals __________

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