CSIR NET Maths previous year question papers

Suppose the distribution of X given $\theta$ is normal with mean $\theta$ and variance 15. Further, let the prior (improper) distribution of $\theta$ be proportional to $1, -\infty < \theta < \infty$. If the observed value of X is 13, then which of the following statements is true?

Suppose we want to estimate the population mean Ȳ of a variable for a finite population of size 85, with 34 Statisticians and 51 Biologists. We consider the following sampling scheme: a stratified random sample with 2 strata of Statisticians (Stratum-1) and Biologists (Stratum-2), where 12 Statisticians and 15 Biologists are drawn from Stratum-1 and Stratum-2, respectively, using SRSWOR scheme. Denote ȳ_S, ȳ_B, and ȳ as the mean of the variable among the Statistician sample, Biologist sample, and the combined sample, respectively. Which of the following is an unbiased estimator of Ȳ?

Let X, Y, and Z be independent Normal random variables with means −1, 0, and 1, respectively, and variances 1, 1, and 3, respectively. Which of the following random variables has a Cauchy distribution with location parameter 0 and scale parameter 1?

Consider an M/M/3 queuing system with arrival rate $\lambda = 2$ and service rate $\mu = 3/2$. Define, for $i = 1, 2, \dots, I_{i} = 1$ if the first transition is from i to i + 1, and $I_{i} = 0$ if the first transition from i is i − 1. Then, $Var(I_{4})$ equals