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:
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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