List of top Statistics Questions on Estimation Theory asked in CUET (PG)

Mean height of plants obtained from a random sample of size 100 is 64 inches. The population standard deviation of the plants is 3 inches. If the plant heights are distributed normally, then the 99% confidence limits of the mean population height of plants, are:

A sample of size 1600 is taken from a population of fathers and sons and the correlation between their heights is found to be 0.80. Then, the correlation limits for the entire population are:

If \(X_1, X_2, \dots, X_n\) is a random sample from the population \(f(x, \theta) = (\theta+1)x^\theta; 0<x<1; \theta>-1\) and \(Y = -\sum_{i=1}^{n} \log(x_i)\). Then \(E\left(\frac{1}{Y}\right)\) is

A random variable X has a distribution with density function
\( f(x;\beta) = \begin{cases} (\beta+1)x^\beta; & 0<x<1; \beta>-1 \\ 0; & \text{otherwise} \end{cases} \) Based on 'n' observations on X, Maximum Likelihood Estimator (MLE) of \(\beta\) is

Let \( \hat{\lambda} \) be the Maximum Likelihood Estimator of the parameter \(\lambda\), then, on the basis of a sample of size 'n' from a population having the probability density function \( f(x; \lambda) = \frac{e^{-\lambda} \lambda^x}{x!} \); \(x = 0, 1, 2, \dots\), \(\lambda>0\), the Var(\(\hat{\lambda}\)) is

Consider the probability density function \( f(x;\theta) = \begin{cases} \frac{2x}{5\theta} & ; 0 \le x \le \theta \\ \frac{2(5-x)}{5(5-\theta)} & ; \theta \le x \le 5 \end{cases} \) For a sample of size 3, let the observations are, \( x_1 = 1, x_2 = 4, x_3 = 2 \). Then, the value of likelihood function at \( \theta=2 \) is

If \(X \sim \beta_1(\alpha, \beta)\) such that parameters \(\alpha, \beta\) are unknown, then the sufficient statistic for \((\alpha, \beta)\) is

If X is a random variable such that,
\(P(X \le x) = \begin{cases} 0 & ; x<0 \\ 1-e^{-x\theta} & ; x \ge 0 \end{cases}\)
based on 'n' independent observations on X, the Maximum Likelihood Estimator (MLE) of E(X) is