Re: Question about normal distributions. **No poker content**
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What is the probability that a randomly selected sample from P1 will be greater than a random sample from P2??
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If the sample is small, it depends on the distributions. If the sample is large, you can use a normal approximation to each average.
The linear combination of two independent normally distributed random variables has a normal distribution. In particular, the difference is normally distributed. You want to know the probability that the difference is greater than 0.
If X and Y are independent and have standard deviations x and y, then the linear combination aX+bY has standard deviation sqrt((ax)^2+(by)^2).
So, as an example, suppose P1 has mean 1 and standard deviation 15. Suppose P2 has mean 0 and standard deviation 20. Supppose you have 100 samples from each with averages A1 and A2. A1 has mean 1 and standard deviation 1.5. A2 has mean 0 and standard deviation 2. A1-A2 has mean 1-0 and standard deviation sqrt((1.5)^2+(2)^2)=sqrt(6.25)=2.5. That A1-A2=0 would be 1/2.5 = 0.4 standard deviations below the mean.
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