Question

Let 2.5, -1.0, 0.5, 1.5 be the observed values of a random sample of size 4 from a continuous distribution with the probability density function
\(f(x)=\frac{1}{8}e^{-|x-2|}+\frac{3}{4\sqrt{2\pi}}e^{-\frac{1}{2}(x-\theta)^2},\ \ x \in \R,\)
where θ ∈ \(\R\) is unknown. Then the method of moments estimate of θ equals __________ (round off to 2 decimal places)

Answer 1

Correct Answer: 0.48

\[\text{Sample mean: }\bar X=\frac{2.5-1.0+0.5+1.5}{4}=0.875.\]

\[f(x)=\frac{1}{8}e^{-|x-2|}+\frac{3}{4\sqrt{2\pi}}e^{-\frac12(x-\theta)^2},\]

\[\mathbb{E}[X]=\frac14(2)+\frac34(\theta)=\frac12+\frac34\theta.\]

\[\frac12+\frac34\theta = 0.875.\]

\[\frac34\theta = 0.375.\]

\[\theta = 0.50.\]

\[\boxed{0.50}\]