Teaching CSIR NET Mock Test Series Mathematical Science Statistics & Exploratory Data Analysis Random Variables & Distribution Functions
Consider a scenario where\( Y_1, Y_2, ..., Y_n \)are independently and identically distributed \(N(μ, φ^{-2}) \)random variables, where \(φ^{-2 }> 0. \)Let the prior distribution on \(φ^2\) have density\( ρ(φ^2) ∝ (1/φ^2)β\) for some β > 0. Which would be correct to say?
1
The prior distribution on φ^2 is an inverse gamma distribution
2
The posterior distribution of \(φ^2\) after observing \(Y_1, ..., Y_n\) is proportional to \((1/φ^2)^{n/2 + β} \times e^{(-nȲ^2/2φ^2) }\)
3
The joint prior distribution of \((μ, φ^2)\) is a normal-inverse gamma distribution when μ has a normal prior distribution with mean 0 and variance \(φ^2\)
4
The posterior mean of\( φ^2 \)is \((nȲ^2 + β)/((n/2) + β - 1)\) when β > 1/2 and n is large E. The joint distribution of \((μ, φ^2)\) is a Pareto distribution.