Order Statistics and Sampling Distributions - Joint Distributions - Statistics Previous Year Questions

Let \(X_1, ..., X_n\) be a random sample from uniform distribution defined over \((0, \theta)\), where \(\theta > 0\) and \(n \geq 2\). Let \(X_{(1)} = \min\{X_1, ..., X_n\}\) and \(X_{(n)} = \max\{X_1, ..., X_n\}\). Then the covariance between \(X_{(n)}\) and \(X_{(1)}/X_{(n)}\) is

Let \(X_1, ..., X_{10}\) be independent and identically distributed normal random variables with mean 0 and variance 2. Then \(E\left(\frac{X_1^2}{X_1^2+\cdots+X_{10}^2}\right)\) is equal to ...

If the marginal probability density function of the $k^{th}$ order statistic of a random sample of size $8$ from a uniform distribution on $[0,2]$ is \(f(x) = \begin{cases} \frac{7}{32} x^6 (2 - x), & 0 < x < 2, \\ 0, & \text{otherwise}, \end{cases}\) then $k$ equals ________

Let X_(1) < X_(2) < X_(3) < X_(4) < X_(5) be the order statistics corresponding to a random sample of size 5 from a uniform distribution on [0, θ], where θ ∈ (0, ∞). Then which of the following statements is/are true?
P : 3X_(2) is an unbiased estimator of θ.
Q : The variance of E[2X_(3) | X_(5)] is less than or equal to the variance of 2X_(3).

Let \( X_1, X_2, ..., X_7 \) be a random sample from a normal population with mean 0 and variance \( \theta > 0 \). Let \[ K = \frac{X_1^2 + X_2^2}{X_1^2 + X_2^2 + \cdots + X_7^2}. \] Consider the following statements: (I) The statistics \( K \) and \( X_1^2 + X_2^2 + \cdots + X_7^2 \) are independent. (II) \( \frac{7K}{2} \) has an \( F \)-distribution with 2 and 7 degrees of freedom. (III) \( E(K^2) = \frac{9}{63} \). Then which of the above statements is/are true?

Suppose a random sample of size 3 is taken from a distribution with the probability density function
\[ f(x) = \begin{cases} 2x, & 0 < x < 1, \\ 0, & \text{elsewhere}. \end{cases} \]
If \(p\) is the probability that the largest sample observation is at least twice the smallest sample observation, then the value of \(p\) (rounded off to three decimal places) is ______

Let \(Y_1 < Y_2 < \cdots < Y_n\) be the order statistics of a random sample of size \(n\) from a continuous distribution, which is symmetric about its mean \(\mu\). Then the smallest value of \(n\) (in integer) such that \(P(Y_1 < \mu < Y_n) \ge 0.99\), is ________