List of top Estimation Questions on Random Variables asked in GATE ST

Let $0,1,1,2,0$ be five observations of a random variable $X$ which follows a Poisson distribution with parameter $\theta>0$. Let the minimum variance unbiased estimate of $P(X\le 1)$, based on this data, be $\alpha$. Then $5^{4}\alpha$ (in integer) is equal to $\;__________$

A random sample of size $5$ is taken from the distribution with density \[ f(x;\theta)= \begin{cases} \dfrac{3x^2}{\theta^3}, & 0[6pt] 0, & \text{elsewhere}, \end{cases} \] where $\theta$ is unknown. If the observations are $3,6,4,7,5$, then the maximum likelihood estimate of the $1/8$ quantile of the distribution (rounded off to one decimal place) is __________.

Consider a gamma distribution with pdf \[ f(x;\beta)= \begin{cases} \dfrac{1}{24\beta^{5}}x^{4}e^{-x/\beta}, & x>0,\\ 0, & \text{elsewhere}, \end{cases} \] with $\beta>0$. For $\beta=2$, find the Cramér–Rao lower bound (rounded to one decimal place) for the variance of any unbiased estimator of $\beta^{2}$ based on a random sample of size 8.

A random sample of size 4 is taken from the distribution with the probability density function \[ f(x; \theta) = \begin{cases} \frac{2(\theta - x)}{\theta^2}, & 0<x<\theta, \\ 0, & \text{elsewhere}. \end{cases} \] If the observed sample values are 6, 5, 3, 6, then the method of moments estimate (in integer) of the parameter $ \theta $, based on these observations, is ________

Let $ X_1, X_2, \dots, X_{18} $ be a random sample from the distribution \[ f(x; \theta) = \begin{cases} \frac{2x}{\theta} e^{-x^2/\theta}, & x>0,\\ 0, & x \leq 0. \end{cases} \] What is the maximum likelihood estimator (MLE) of $ \theta $?

Let $ X_1, X_2, \dots, X_n $ be a random sample from a population $ f(x; \theta) $, where $ \theta $ is a parameter. Then which one of the following statements is NOT true?

Let $ S^2 $ be the variance of a random sample of size $ n>1 $ from a normal population with an unknown mean $ \mu $ and an unknown finite variance $ \sigma^2>0 $. Consider the following statements:
(I) $ S^2 $ is an unbiased estimator of $ \sigma^2 $, and $ S $ is an unbiased estimator of $ \sigma $.
(II) $ \frac{n-1}{n} S^2 $ is a maximum likelihood estimator of $ \sigma^2 $, and $ \sqrt{\frac{n-1}{n}} S $ is a maximum likelihood estimator of $ \sigma $.
Which of the above statements is/are true?