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.

Question

UpStudy AI · Accepted Answer

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.
The distribution of $$ z_{1}^{2} - z_{2}^{2} $$ is the difference of two independent chi-square distributions with 1 degree of freedom each. The mean of this distribution is 0.

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.

Solución de tutoría real

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Statistics Questions