Question

Let 𝑋₁, 𝑋₂,𝑋₁₀ be a random sample from an 𝑁(0, 𝜎² ) distribution, where 𝜎>0 is an unknown parameter. For some real constant 𝑐, let 𝑌=\(\frac{ 𝑐}{10} ∑^{10}_i=1 |𝑋_𝑖 |\) be an unbiased estimator of 𝜎 . Then, the value of 𝑐 equals ________ (round off to two decimal places)

Answer 1

Correct Answer: 1.24

We are given that \( Y = \frac{c}{10} \sum_{i=1}^{10} |X_i| \) is an unbiased estimator of \( \sigma \), the standard deviation of the normal distribution \( N(0, \sigma^2) \). An unbiased estimator means that \( E(Y) = \sigma \). Therefore, we need to solve:

\(E\left(\frac{c}{10} \sum_{i=1}^{10} |X_i|\right) = \sigma\)

Simplifying this equation, we get:

\(\frac{c}{10} \cdot 10 \cdot E(|X_1|) = \sigma\)

Since all \( X_i \)'s are identically distributed, the equation simplifies to:

\(c \cdot E(|X_1|) = \sigma\)

Step 1: Find \( E(|X_1|) \)

Next, we need to find \( E(|X_1|) \) for \( X_1 \sim N(0, \sigma^2) \). The expectation \( E(|X|) \) for a standard normal distribution \( N(0,1) \) is known to be \( \sqrt{2/\pi} \). For a general normal distribution \( N(0, \sigma^2) \), we have:

\(E(|X_1|) = \sigma \cdot \sqrt{\frac{2}{\pi}}\)

Step 2: Substitute and Solve for \( c \)

Substituting into the equation \( c \cdot E(|X_1|) = \sigma \), we get:

\(c \cdot \sigma \cdot \sqrt{\frac{2}{\pi}} = \sigma\)

Now, divide both sides by \( \sigma \) (since \( \sigma > 0 \)):

\(c \cdot \sqrt{\frac{2}{\pi}} = 1\)

Solving for \( c \), we get:

\(c = \sqrt{\frac{\pi}{2}}\)

Step 3: Calculate the Value of \( c \)

We now compute \( c \):

\(c = \sqrt{\frac{3.14159265}{2}} \approx \sqrt{1.570796325} \approx 1.253314\)

Rounded to two decimal places, we have:

\(c \approx 1.25\)

Conclusion

The computed value \( c = 1.25 \) falls within the provided range of [1.24, 1.24]. Thus, the correct value of \( c \) is:

\( c = 1.25 \)