# Difference between two normal distributions

#### gralla55

I have two normal distributions, with different means but the same variance. How can I find the probability for that the difference between a sample taken from each distribution is above x? Not sure how to approach this. Would appreciate any help.

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#### matheagle

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$$\displaystyle P(\bar X-\bar Y>a)=P\left( {(\bar X-\bar Y)-(\mu_X-\mu_Y)\over \sqrt{ {\sigma^2\over n} + {\sigma^2\over m}}}>{a-(\mu_X-\mu_Y)\over \sqrt{ {\sigma^2\over n} + {\sigma^2\over m}}}\right)$$

$$\displaystyle =P\left( Z>{a-(\mu_X-\mu_Y)\over \sqrt{ {\sigma^2\over n} + {\sigma^2\over m}}}\right)$$