Question

Let 𝑋 and 𝑌 be i.i.d. random variables each having the 𝑁(0, 1) distribution. Let 𝑈=\(\frac{𝑋}{𝑌}\) and 𝑍=|𝑈|. Then, which of the following statements is/are TRUE?

• A. 𝑈 has a Cauchy distribution
• B. 𝐸(𝑍^𝑝)<∞, for some 𝑝 ≥ 1
• C. 𝐸(𝑒^𝑡𝑍) does not exist for all 𝑡∈(−∞, 0)
• D. 𝑍 ²~𝐹_1,1

Answer 1

The Correct Option is A, D

To solve the given problem, we need to analyze the distributional properties of the random variables 𝑈 and 𝑍. 

1. Consider the random variable \(𝑈 = \frac{𝑋}{𝑌}\). Since both 𝑋 and 𝑌 are independent and identically distributed (i.i.d.) standard normal random variables, 𝑈 follows a Cauchy distribution with location parameter 0 and scale parameter 1. This is a well-known result in probability theory when dealing with the ratio of two standard normal variables. Therefore, the statement "𝑈 has a Cauchy distribution" is true.
2. Next, consider the random variable \(𝑍 = |𝑈|\). Since 𝑈 is Cauchy distributed, 𝑍 is distributed as half-Cauchy.
3. Now, examine the statement "𝐸(𝑍^{𝑝}) < ∞, for some 𝑝 ≥ 1". The Cauchy distribution does not have a finite expectation or higher-order moments. Since the expectation of \(𝑈\) is undefined, so are the moments of \(𝑍\). Therefore, this statement is false.
4. Check the statement "𝐸(𝑒^{𝑡𝑍}) does not exist for all 𝑡 ∈ (−∞, 0)". This statement is false because, for a Cauchy distribution, the moment generating function does not exist, making it impossible to find an expected value for an exponential function involving 𝑍.
5. Finally, let's consider the variable \(𝑍^2\). The statement "𝑍^2 ~ F_{1,1}" is true because, if 𝑈 has a standard Cauchy distribution, then \(𝑍^2 = 𝑈^2\) follows an \(F\)-distribution with parameters \((1,1)\).

Thus, the correct statements are:

• 𝑈 has a Cauchy distribution.
• \(𝑍^2 \sim F_{1,1}\).