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This might seem very simple but I'm having some trouble getting to the answer. If I have a random variable that's normally distributed $$X\sim N(30, 3^2)$$ and another random var. $$Y \sim N(20, (2.5)^2) $$ and I want to find the $P(2X+5Y<175)$ how would I go about doing this?

*They are both independent

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    $\begingroup$ you need to know whether $X$ and $Y$ are independent. then you should use the fact that the sum of two independent normals is again normal. $\endgroup$
    – symplectomorphic
    Apr 30, 2014 at 3:04
  • $\begingroup$ sorry, yes they are $\endgroup$
    – user1530491
    Apr 30, 2014 at 3:06

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  1. Compute the expectation of $2X + 5Y$.
  2. Compute the variance and standard deviation of $2X + 5Y$.
  3. Express the distribution of $2X + 5Y$ as a normal random variable $J$.
  4. Express the probability $ P(J<175)$ in terms of the standardized random variable $Z$.
  5. Look up the answer in a $Z$-table.
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