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I am new to this site so I'm not quite sure how this whole thing goes. Anyway, I have a question from my homework for statistics class. I'm new to statistics so questions that seem easy to you might be a little bit more difficult for me. I'm having one of those moments where my mind has gone blank. The problem that I'm stuck at is this:

Suppose that $X_1$ and $X_2$ are jointly normally distributed & that $E[X_1] = 1,E[X_2] = 2,var[X_1] = 3,var[X_2] = 4,cov[X_1, X_2] = 1$.

How do you find the distribution of $Y= 1 + 2X_1 + 3X_2$? Thank you ahead of time to those who'll explain this problem to me. I really appreciate your time. Do you treat it kind of like a function and plug in the first values for expected mean. Then do you do the same for variance?

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2 Answers 2

A linear combination of jointly distributed normal random variables is normal. If $Y$ is normal, then it has two parameters, the mean and the variance. Now, all you have to do is compute these two quantities to get the answer.

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This is a problem of Bi-variate Normal Distribution. The marginal distributions of a Bi-variate Normal Distribution are uni-variate normals.

So here $ X_1 \sim \mathcal{N}(1,3) \, \text{and} \, X_2 \sim \mathcal{N}(2,4)$.

Again $Y$ being a linear combination of two normals, is also follow a normal distribution with mean $1+2E(X_1)+3E(X_2)$ and variance $Var(1+2X_1+3X_2)$. Now compute the variance. But be careful because $X_1$ and $X_2$ are not independent.

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