Suppose that X1, X2, X3, X4 are a random sample from a N (0, σ^2) distribution. Let U = X1 + X2 + X3 and V = [ ( √3 ) * X4 ] / U. Give the names of the following distributions as well as the value(s) of their parameter(s): (i) U / 3. (ii) (U^2)/(3*σ^2). (iii) X4 / σ. (iv) V.

Let's break down each part of the problem carefully, identifying the distributions and their parameters.

Given that \( X_1, X_2, X_3, X_4 \) are independent and identically distributed (i.i.d.) random variables from a normal distribution \( N(0, \sigma^2) \), we can derive the distributions of the variables asked for:

### (i) \( \frac{U}{3} \)

- \( U = X_1 + X_2 + X_3 \), and each \( X_i \sim N(0, \sigma^2) \).
- Since \( U \) is the sum of 3 independent normal variables, \( U \sim N(0, 3\sigma^2) \).
- Dividing by 3 gives:
  \[
  \frac{U}{3} \sim N\left( 0, \frac{3\sigma^2}{9} \right) = N\left( 0, \frac{\sigma^2}{3} \right)
  \]
  **Answer**: \( \frac{U}{3} \sim N\left(0, \frac{\sigma^2}{3}\right) \).

### (ii) \( \frac{U^2}{3\sigma^2} \)

- \( U \sim N(0, 3\sigma^2) \), meaning \( \frac{U}{\sqrt{3}\sigma} \sim N(0, 1) \).
- Squaring this gives a chi-square distribution with 1 degree of freedom:
  \[
  \left( \frac{U}{\sqrt{3}\sigma} \right)^2 \sim \chi^2_1
  \]
  Hence:
  \[
  \frac{U^2}{3\sigma^2} \sim \chi^2_1
  \]
  **Answer**: \( \frac{U^2}{3\sigma^2} \sim \chi^2_1 \) (chi-square distribution with 1 degree of freedom).

### (iii) \( \frac{X_4}{\sigma} \)

- \( X_4 \sim N(0, \sigma^2) \).
- Dividing by \( \sigma \) gives a standard normal distribution:
  \[
  \frac{X_4}{\sigma} \sim N(0, 1)
  \]
  **Answer**: \( \frac{X_4}{\sigma} \sim N(0, 1) \) (standard normal distribution).

### (iv) \( V = \frac{\sqrt{3} X_4}{U} \)

- From (i), \( U \sim N(0, 3\sigma^2) \).
- Also, \( X_4 \sim N(0, \sigma^2) \), so \( \frac{X_4}{\sigma} \sim N(0, 1) \).
- We now have a ratio of a normal variable \( \sqrt{3} X_4 \sim N(0, 3\sigma^2) \) to another normal variable \( U \sim N(0, 3\sigma^2) \).
- The ratio of two independent normal distributions with mean 0 follows a **Cauchy distribution**:
  \[
  V = \frac{\sqrt{3} X_4}{U} \sim C(0, 1)
  \]
  **Answer**: \( V \sim C(0, 1) \) (standard Cauchy distribution).

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### Summary of Distributions:

1. \( \frac{U}{3} \sim N\left( 0, \frac{\sigma^2}{3} \right) \).
2. \( \frac{U^2}{3\sigma^2} \sim \chi^2_1 \).
3. \( \frac{X_4}{\sigma} \sim N(0, 1) \).
4. \( V = \frac{\sqrt{3} X_4}{U} \sim C(0, 1) \).

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Would you like more details on any of the steps?

Here are 5 related questions to extend the problem:

1. How would the distribution of \( U \) change if the sample came from a \( N(\mu, \sigma^2) \) distribution instead?
2. What is the distribution of \( \frac{X_1 + X_2}{2} \)?
3. Can you derive the moment-generating function for the distribution of \( \frac{U^2}{3\sigma^2} \)?
4. What is the cumulative distribution function of the standard Cauchy distribution?
5. How would the distribution of \( V \) change if the numerator was \( X_3 + X_4 \)?

**Tip**: When dealing with sums of independent normal variables, always remember that their sum is also normally distributed, with the variance equal to the sum of the individual variances.

Suppose that X1, X2, X3, X4 are a random sample from a N(0, σ^2) distribution. Let U = X1 + X2 + X3 and V = [(√3) * X4] / U. Give the names of the following distributions as well as the value(s) of their parameter(s): (i) U / 3. (ii) (U^2)/(3*σ^2). (iii) X4 / σ. (iv) V.

Explore the distributions of U = X1 + X2 + X3 and V = [(√3) * X4] / U derived from random variables sampled from a N(0, σ²) distribution. This solution covers normal, chi-square, and Cauchy distributions, including their parameters. Suitable for undergraduate students studying probability and statistics.

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