I'm developing some game theory models that involve normal distributions, and am not sure how to solve this probability problem:

Suppose there are two normal distributions:

$$

X_1 sim N(mu_1, sigma^2)

$$

$$

X_2 sim N(mu_2, sigma^2)

$$

You know what both distributions are, but that's all you know. I take a draw (call it $s$) from one of the two distributions and show it to you. Given $s$, what is the probability I chose from $X_1$?

Thanks for any help!

Here's of a picture of two overlapping normal distributions in case it's helpful to have a visual:

**Contents**hide

#### Best Answer

As Adrian already suggested you need to know the prior probability that $X$ came from each distribution. If $Y$ is an indicator telling us whether or not $X$ came from distribution one and $p_1$ and $p_2$ are the mixing (prior) probabilities then

$$ P(Y = 1 mid X = x) = frac{p_1 f_{X mid Y=1}(x)}{p_1 f_{X mid Y=1}(x) + p_2 f_{X mid Y=0}(x)} . $$

All you've specified are the conditional densities $f_{X mid Y = 1}$ and $f_{X mid Y = 0}$ but this isn't enough to calculate the probability. You also need to know something about $p_1$ and $p_2$.

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