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For this problem, we are told that $X$ and $Y$ are jointly normally distributed variables, both being standard normal. We're given their correlation coefficient. So, how do I get from there to finding the probability that $X$ + $Y$ $<$ 0.7?

What I've worked out so far is that the $\operatorname{Cov}(X, Y)$ equals the correlation coefficient since $X$ and $Y$ are standard normal (because the standard deviations are $1$), which is also equal the $E[XY]$ (because the means are $0$). But where do I go from here...?

user642796
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Shell
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1 Answers1

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Hint: $X+Y$ is a normal random variable since $X$ and $Y$ are jointly normal. Work out its mean and variance and look up the answer in tables.

Dilip Sarwate
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  • Okay... So I get that the mean is then 0, and the standard deviation is the square root of 2. But what tables are you referring to? – Shell Apr 17 '14 at 18:57
  • If you have been taught anything at all about normal random variables before being assigned this problem, you would have been told about the cumulative probability distribution function of the standard normal random variable, and tables of the values of this function. You might even have been taught how to use the table for other (nonstandard) normal random variables such as $X+Y$ here, but if not, read this answer for a detailed description of how to use the table. – Dilip Sarwate Apr 17 '14 at 19:05