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.

If and are two independent S.N.Vs then name the distribution of
and write its mean.

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Statistics Questions

If and are two independent S.N.Vs then name the distribution of and write its mean.