Solution Found!
Let Y1, Y2, . . . , Yn be independent, normal random
Chapter 6, Problem 6.41(choose chapter or problem)
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: