SOLUTION: 3 7. Let X1, X2, X3, X4 be a random sample of size 4 from a standard normal population. Find the sampling distribution (if possible) and moment gener- ating function of the statist

Algebra ->  Probability-and-statistics -> SOLUTION: 3 7. Let X1, X2, X3, X4 be a random sample of size 4 from a standard normal population. Find the sampling distribution (if possible) and moment gener- ating function of the statist      Log On


   

Question 1164808: 3 7. Let X1, X2, X3, X4 be a random sample of size 4 from a standard normal population. Find the sampling distribution (if possible) and moment gener- ating function of the statistic 2X?+3X2 +X +4X. What is the probability distribution of the sample?
Answer by CPhill(2189) About Me  (Show Source):
You can put this solution on YOUR website!
To find the sampling distribution and the moment-generating function (MGF) of the given statistic, we rely on the properties of independent normal random variables.
### 1. The Probability Distribution of the Sample
Since is a random sample of size 4 from a **standard normal population**, each is independent and identically distributed (i.i.d.) such that:


The joint probability distribution of the sample is the product of their individual normal density functions:

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### 2. The Sampling Distribution of the Statistic
Let the statistic be .
Note that for any , the square of the variable follows a **Chi-square distribution with 1 degree of freedom**:

However, our statistic is a **weighted sum** of independent Chi-square variables:

Because the coefficients (weights) are not all equal to 1, does **not** follow a standard Chi-square distribution. Instead, it follows a **Generalized Chi-square distribution** (specifically, a linear combination of Chi-squares). There is no simple named probability density function for this, but it is fully characterized by its MGF.
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### 3. Moment-Generating Function (MGF)
The MGF of a standard Chi-square variable is:

Since the variables are independent, the MGF of a sum is the product of the individual MGFs. For a weighted variable , the MGF is .

Substitute the formula:

**Condition for Existence:** The MGF is defined only when all terms inside the square root are positive. The strictest constraint comes from , so .
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### Summary
* **Sample Distribution:** Jointly Normal .
* **Statistic Distribution:** A weighted sum of variables (Generalized Chi-square).
* **MGF:** for .
Would you like me to calculate the **mean** and **variance** of this statistic using its MGF?