# JAM 2022 [ 41 -50]

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 funciton

f(x) = \frac{1}{8} e^{-|x-2|} + \frac{3}{4\sqrt{2\pi}}e^{-\frac{1}{2}(x-\theta)^2}, \quad x \in \mathbb{R},

where \theta \in \mathbb{R} is unknown. Then the method of moments estimate of \theta equals ________________ (round off to 2 decimal places)

Noting that there appears to be the double exponential /Laplace distribution and normal distribution pdf forms in the given pdf.

E(X) = \frac{2}{4} + \frac{3\theta}{4} \\
\text{sample mean, } \bar{x} = 0.875 \\
0.5 + \frac{3\hat{\theta}}{4} =0.875 \\
\implies \frac{3\hat{\theta}}{4} = 0.375 \\
\implies \hat{\theta} = 0.5