Question 1164809: let x1,x2,x3,x4 be a random sample of size 4 from a standard normal population.find the distribution of the statistics x1+x2/square root(x2 square +3x square)
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! To find the distribution of the statistic , we first need to evaluate the properties of the numerator and denominator based on the fact that are i.i.d. .
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### 1. Analyze the Numerator ()
Let .
Since and are independent normal variables:
* **Mean:**
* **Variance:**
Therefore, . We can standardize this by noting that .
### 2. Analyze the Denominator
There appears to be a slight typo in your expression ( square). Assuming the statistic is meant to be , or a similar variation involving independent variables, let's look at the standard form of a **Student's t-distribution**.
A t-distribution is formed by:
where , , and and are independent.
### 3. Re-evaluating the specific statistic
If the statistic is exactly :
1. **Numerator:** , where .
2. **Denominator:** is the sum of the squares of 2 independent standard normal variables, so .
Substituting these into :
Divide both the numerator and denominator by (the square root of the degrees of freedom ):
### 4. Conclusion
By definition, follows a **Student's t-distribution with 2 degrees of freedom**.
**The distribution of the statistic is .**
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### If the statistic was specifically:
This would be more complex because appears in both the numerator and denominator, meaning they are **not independent**. In such cases, the distribution would not follow a standard t-distribution.
However, in most textbook problems of this type, the variables in the denominator are independent of those in the numerator. Would you like me to show the derivation for a different combination of these variables?
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