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.