### Solution Found!

# Let Y1, Y2, . . . , Yn be independent, normal random

**Chapter 6, Problem 6.41**

(choose chapter or problem)

**QUESTION:**

Let \(Y_{1}, Y_{2^{\prime} \ldots}, Y_{n}\) be independent, normal random variables, each with mean \(\mu\) and variance \(\sigma^{2}\). Let \(a_{1}, a_{2}, \ldots, a_{n}\) denote known constants. Find the density function of the linear combination \(U=\sum_{i=1}^{n} a_{i} Y_{i}\).

### Questions & Answers

**QUESTION:**

Let \(Y_{1}, Y_{2^{\prime} \ldots}, Y_{n}\) be independent, normal random variables, each with mean \(\mu\) and variance \(\sigma^{2}\). Let \(a_{1}, a_{2}, \ldots, a_{n}\) denote known constants. Find the density function of the linear combination \(U=\sum_{i=1}^{n} a_{i} Y_{i}\).

**ANSWER:**

Step1 of 2:

We know that the definition of moment generating function of random variable Y is:

\(m_{Y}(t)=E\left(e^{t Y}\right)\)

Now, moment generating function of normally distributed random variable is:

\(m_{Y_{i}}(t)=E\left(e^{\frac{t \mu+\sigma^{2} t^{2}}{2}}\right)\)

Step2 of 2: