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

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) (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:

---
### 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.
---
### 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 .
---
### 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?
RELATED QUESTIONS
let x1,x2,x3,x4 be a random sample of size 4 from a standard normal population.find the... (answered by CPhill)

Let P(x) be the probability function of a discrete random variable x which assumes the... (answered by stanbon)

Let X1 represent the mean of a sample of size n, = 2, selected with replacement, from the (answered by proyaop)

a) If a random sample of size n = 6 is taken from a population, what is required in order (answered by Boreal)

i have to face exam next monday.before it toget confidence i ant to know this ones... (answered by robertb)

Suppose (X1,X2,…,X9) is a random sample from Normal(2,42). Let X(bar) be the sample... (answered by CPhill)

A random sample of size 20 drawn from a normal distribution has a sample mean of 8 and a... (answered by stanbon)

Suppose a simple random sample of size n = 40 is taken from a population with μ=50... (answered by Boreal)

Let X1, X2, X3, ...,X 50 be a random sample of size 50 from a distribution with density... (answered by CPhill)