24) If \( z_{1} \) and \( z_{2} \) are two independent S.N.Vs then name the distribution of \( \left(z_{1}^{2}-z_{2}^{2}\right) \) and write its mean.

If \( z_{1} \) and \( z_{2} \) are two independent standard normal variables (S.N.Vs), then \( z_{1}^{2} \) and \( z_{2}^{2} \) each follow a chi-square distribution with 1 degree of freedom, denoted as \( \chi^2(1) \). ### Distribution of \( z_{1}^{2} - z_{2}^{2} \) The difference \( z_{1}^{2} - z_{2}^{2} \) represents the difference between two independent chi-square distributed random variables, each with 1 degree of freedom. This type of distribution does not correspond to a standard named distribution like the normal, chi-square, or F-distribution. Instead, it is characterized as the **difference of two independent chi-square distributions with 1 degree of freedom each**. Mathematically, if: - \( X = z_{1}^{2} \sim \chi^2(1) \) - \( Y = z_{2}^{2} \sim \chi^2(1) \) Then: \[ W = X - Y \] is the difference of two independent chi-square(1) variables. ### Mean of \( z_{1}^{2} - z_{2}^{2} \) To find the mean of \( W = z_{1}^{2} - z_{2}^{2} \): \[ \mathbb{E}[W] = \mathbb{E}[z_{1}^{2}] - \mathbb{E}[z_{2}^{2}] \] Since both \( z_{1}^{2} \) and \( z_{2}^{2} \) have an expected value of 1 (because the mean of a \( \chi^2(1) \) distribution is equal to its degrees of freedom, which is 1): \[ \mathbb{E}[W] = 1 - 1 = 0 \] ### Summary - **Distribution:** \( z_{1}^{2} - z_{2}^{2} \) is the difference of two independent chi-square distributions, each with 1 degree of freedom. - **Mean:** The mean of \( z_{1}^{2} - z_{2}^{2} \) is **0**. **Answer:** Its distribution is equal to the difference of two independent chi-square (1) variables, and its mean is zero.