Question

Let X₍₁₎ < X₍₂₎ < ⋯ < X₍₉₎ be the order statistics corresponding to a random sample of size 9 from U(0, 1) distribution. Then which one of the following statements is NOT true ?

• A. \(E(\frac{X_{(9)}}{1-X_{(9)}})\) is finite
• B. E(X₍₅₎) = 0.5
• C. The median of X₍₅₎ is 0.5
• D. The mode of X₍₅₎ is 0.5

Answer 1

The Correct Option is A

To solve this problem, we need to evaluate the truthfulness of each given statement about the order statistics from a sample of size 9 from a uniform distribution \( U(0, 1) \):

1. \(E\left(\frac{X_{(9)}}{1-X_{(9)}}\right)\) is finite:
   • The random variable \( X_{(9)} \) represents the maximum of the 9 random variables drawn from \( U(0,1) \).
   • The distribution for \( X_{(9)} \) is such that its cumulative distribution function (CDF) is \( F_{X_{(9)}}(x) = x^9 \).
   • The expected value \( E\left(\frac{X_{(9)}}{1-X_{(9)}}\right) \) is formulated as an integral: \[ E\left(\frac{X_{(9)}}{1-X_{(9)}}\right) = \int_0^1 \frac{x}{1-x} \cdot 9x^8 \, dx \]
   • This involves a singularity at \( x = 1 \), causing the expected value to be infinite.

   Therefore, the statement "\(E\left(\frac{X_{(9)}}{1-X_{(9)}}\right)\) is finite" is NOT true.

2. \( E(X_{(5)}) = 0.5 \):
   • The expected value of the \( k^{th} \) order statistic \( X_{(k)} \) from a uniform distribution \( U(0, 1) \) is given by \(\frac{k}{n+1}\), where \( n \) is the sample size.
   • For \( X_{(5)} \), this becomes \( \frac{5}{10} = 0.5 \).

   This statement is true.

3. The median of \( X_{(5)} \) is 0.5:
   • The median generally equals the expected value for symmetric distributions such as \( U(0,1) \).
   • Thus, the median is also 0.5.

   This statement is true.

4. The mode of \( X_{(5)} \) is 0.5:
   • The mode of the uniform distribution is not at a particular point and for order statistics, it is not straightforward to calculate directly.
   • However, with symmetric properties and absence of skew in \( U(0,1) \), it might heuristically reflect towards the mean.

   A strict analytical solution for the mode is complex, but for practical purposes in uniform distributions, this can be assumed true or likely without complex derivations.

Hence, the correct answer is that the statement "\(E\left(\frac{X_{(9)}}{1-X_{(9)}}\right)\) is finite" is NOT true, as its value tends to infinity.