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Using the normal distribution. Let $X\sim \mathcal{N}(1, 2)$ and $Y\sim \mathcal{N}(2, 3)$ where $\mathcal{N}(μ, \sigma^2$) denotes the normal distribution with mean $\mu$ and variance $\sigma^2$. X and Y are independent. Let $U = 2X + 3Y$.

What is the mean of U? What is the variance of U? What is $P(6<=U<=7.5)$?

For the mean of U, I calculated $2(1)+3(2)=8$. For the variance of U I calculated $ 2(2)+3(3)=13$.

Then I found pnorm(7.5,mean=8,sd=sqrt13)-pnorm(6,mean=8,sd=sqrt13) [1] 0.1553036 for P(6<=U<=7.5).

Did I do this correctly? Thanks in advance!

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  • $\begingroup$ I vote to close this as a duplicate. Did you read @Corone's edited answer to your other question? $\endgroup$
    – Dilip Sarwate
    Commented Feb 21, 2013 at 23:11
  • $\begingroup$ Yes, I did. But this question has the U=2x+3Y involved. Also, I did not get a confirmation that my 67% was correct? If it is possible, could you verify that I took the correct course of action? $\endgroup$
    – dataznkid1
    Commented Feb 22, 2013 at 0:12

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