List of top Statistics Questions on Estimation asked in IIT JAM MS

Let \(X_1,X_2\) be a random sample of size \(2\) from a distribution with the probability density function
Let \(X_1,X_2,X_3,X_4\) be a random sample of size \(4\) from a distribution with the probability density function
Let \(Y_1,Y_2\) be a random sample of size \(2\) from a distribution with the probability mass function
Let \(X_1,X_2\) be a random sample of size \(2\) from an \(Exp(\theta)\) distribution, where \(\theta>0\) is an unknown parameter. Let
Let \(X_1,X_2,\ldots,X_n\) be a random sample of size \(n\;(n\geq 1)\) from a distribution with the probability density function
Let \(X_1,X_2,\ldots,X_n\) be a random sample of size \(n\ (n\geq2)\) from a \(N(0,\sigma^2)\) distribution, where \(\sigma>0\) is an unknown parameter. Which one of the following statements is true?
Let \( X_1, X_2, \ldots, X_{10} \) be a random sample from a \( U(-\theta, \theta) \) distribution, where \( \theta \in (0, \infty) \). Let \( X_{(10)} = \max \{ X_1, X_2, \ldots, X_{10} \} \) and \( X_{(1)} = \min \{ X_1, X_2, \ldots, X_{10} \} \). If the observed values of \( X_{(10)} \) and \( X_{(1)} \) are 8 and -10, respectively, then the maximum likelihood estimate of \( \theta \) is equal to __________ (answer in integer).
Suppose that the lifetimes (in months) of bulbs manufactured by a company have an \( \text{Exp}(\lambda) \) distribution, where \( \lambda > 0 \). A random sample of size 10 taken from the bulbs manufactured by the company yields the sample mean lifetime \( \bar{x} = 3.52 \) months. Then the uniformly minimum variance unbiased estimate of \( \frac{1}{\lambda} \) based on this sample is equal to __________ months (round off to 2 decimal places).
A biased coin, with probability of head as \( p \), is tossed \( m \) times independently. It is known that \( p \in \left\{ \frac{1}{4}, \frac{3}{4} \right\} \) and \( m \in \{3, 5\}. \) If 3 heads are observed in these \( m \) tosses, then which of the following statements is correct?
A year is chosen at random from the set of years {2012, 2013, … , 2021}. From the chosen year, a month is chosen at random and from the chosen month, a day is chosen at random. Given that the chosen day is the 29th of a month, the conditional probability that the chosen month is February equals
In a store, the daily demand for milk (in litres) is a random variable having Exp(λ) distribution, where λ > 0. At the beginning of the day, the store purchases c (> 0) litres of milk at a fixed price b (> 0) per litre. The milk is then sold to the customers at a fixed price s (> b) per litre. At the end of the day, the unsold milk is discarded. Then the value of c that maximizes the expected net profit for the store equals
Let 0.2, 1.2, 1.4, 0.3, 0.9, 0.7 be the observed values of a random sample of size 6 from a continuous distribution with the probability density function
\(f(x) = \begin{cases}    1, & 0< x \le \frac{1}{2} \\     \frac{1}{2\theta-1}, & \frac{1}{2} \lt x \le \theta \\ 0, & \text{otherwise,}\end{cases}\)
where θ > \(\frac{1}{2}\) is unknown. Then the maximum likelihood estimate and the method of moments estimate of θ, respectively, are
Let X1, X2, X3, X4 be a random sample from a continuous distribution with the probability density function f(x) = \(\frac{1}{2}e^{-|x-\theta|}\), x ∈ \(\R\), where θ ∈ \(\R\) is unknown. Let the corresponding order statistics be denoted by X(1) < X(2) < X(3) < X(4). Then which of the following statements is/are true ?
Consider the linear regression model yi = β0 + β1xi + ∈i , i = 1, 2, … , 6, where β0 and β1 are unknown parameters and ∈i ’s are independent and identically distributed random variables having N(0, 1) distribution. The data on (xi, yi) are given in the following table.
xi1.02.02.53.03.54.5
yi2.03.03.54.25.05.4
If \(\hat{\beta}_0\) and \(\hat{\beta}_1\) are the least squares estimates of β0 and β1, respectively, based on the above data, then \(\hat{β}0 + \hat{β}1\) equals __________ (round off to 2 decimal places)
Let \( X_1, X_2, \dots, X_n \ (n \ge 2) \) be a random sample from a distribution with probability density function \[ f(x; \theta) = \begin{cases} \dfrac{1}{2\theta}, & -\theta \le x \le \theta, \\ 0, & |x| > \theta, \end{cases} \] where \( \theta \in (0, \infty) \) is unknown. If \( R = \min\{X_1, X_2, \dots, X_n\} \) and \( S = \max\{X_1, X_2, \dots, X_n\} \), then which of the following statements is/are TRUE?
Let \( X_1, X_2, \dots, X_n \ (n \ge 2) \) be a random sample from a distribution with probability density function \[ f(x; \theta) = \begin{cases} \dfrac{3x^2}{\theta} e^{-x^3 / \theta}, & x > 0, \\ 0, & \text{otherwise,} \end{cases} \] where \( \theta \in (0, \infty) \) is unknown. If \( T = \sum_{i=1}^n X_i^3 \), then which of the following statements is/are TRUE?
Let \( X_1, X_2, \dots, X_n \ (n \ge 2) \) be a random sample from a distribution with probability density function \[ f(x; \theta) = \begin{cases} \theta x^{\theta - 1}, & 0 \le x \le 1, \\ 0, & \text{otherwise,} \end{cases} \] where \( \theta \in (0, \infty) \) is unknown. Then, which of the following statements is/are TRUE?
Let \( 5, 10, 4, 15, 6 \) be an observed random sample of size 5 from a distribution with probability density function \[ f(x; \theta) = \begin{cases} e^{-(x - \theta)}, & x \ge \theta \\ 0, & \text{otherwise} \end{cases} \] where \( \theta \in (-\infty, 3] \) is unknown. Then, the maximum likelihood estimate (MLE) of \( \theta \) based on the observed sample is equal to ..............
Let \( X_1, X_2, ..., X_n \) (\( n \ge 3 \)) be a random sample from \( \text{Poisson}(\theta) \), where \( \theta>0 \) is unknown, and let \( T = \sum_{i=1}^{n} X_i \). Then, the uniformly minimum variance unbiased estimator (UMVUE) of \( e^{-2\theta}\theta^3 \) is:
Let \( X_1, X_2, ..., X_n \) (\( n \ge 2 \)) be a random sample from \( U(\theta - 5, \theta + 5) \), where \( \theta \in (0, \infty) \) is unknown. Let \( T = \max(X_1, ..., X_n) \) and \( U = \min(X_1, ..., X_n) \). Then, which of the following statements is TRUE?

Let the sample mean based on a random sample from Exp($\lambda$) distribution be 3.7. Then the maximum likelihood estimate of $1 - e^{-\lambda}$ equals ........... (round off to two decimal places).
 

Let $x_1 = 1, x_2 = 0, x_3 = 0, x_4 = 1, x_5 = 0, x_6 = 1$ be the data on a random sample of size 6 from Bin(1, $\theta$) distribution, where $\theta \in (0, 1)$. Then the uniformly minimum variance unbiased estimate of $\theta(1 + \theta)$ equals .............

Let $x_1 = 1, x_2 = 4$ be the data on a random sample of size 2 from a Poisson($\theta$) distribution, where $\theta \in (0, \infty)$. Let $\hat{\psi}$ be the uniformly minimum variance unbiased estimate of $\psi(\theta) = \sum_{k=4}^{\infty} e^{-\theta} \dfrac{\theta^k}{k!}$ based on the given data. Then $\hat{\psi}$ equals ............ (round off to two decimal places). 
 

Let $X_1, X_2, \ldots, X_n$ be i.i.d. Poisson($\lambda$) random variables, where $\lambda > 0$. Define \[ \bar{X} = \frac{1}{n}\sum_{i=1}^n X_i \,\, \text{and} \,\, S^2 = \frac{1}{n - 1}\sum_{i=1}^n (X_i - \bar{X})^2. \] Then which of the following statements is/are TRUE? 
 

Let $X_1, X_2, \ldots, X_n$ be a random sample from $\text{Exp}(\theta)$ distribution, where $\theta \in (0, \infty)$. If $\bar{X} = \dfrac{1}{n}\sum_{i=1}^n X_i$, then a 95% confidence interval for $\theta$ is