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Imagine that I take two separate measures and I get two separate normal distributions $\mathcal{N}_1(m_1, s_1^2)$ and $\mathcal{N}_2(m_2, s_2^2)$

  1. How can I find a single normal distribution $\mathcal{N}_3$ which is a type of average of these two normal curves?

  2. As an extension, how could I find a distribution which represents both normals? Meaning, it would be a single distribution and it would have two peaks and would be centered at $\frac{m_1 + m_2}{2}$?

Arpit Sisodia
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CodeGuy
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    For the first question, why not just average the two curves? If $X_1$ and $X_2$ are your two normal distributions, define $X_3$ as the average of $X_1$ and $X_2$: $X_3=(X_1+X_2)/2$. $X_3$ will follow a normal distribution with mean $(m_1+m_2)/2$ and variance $(s_1^2+s_2^2)/4$. For the second question, look into mixture models. Rather than averaging the distributions, you will define a new density function as the average of the original density functions. – assumednormal Oct 27 '12 at 02:42
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    How do you get two normal distributions? Does each measurement consist of many observations (samples) from which you estimate the values of $m_1$ and $s_1^2$ as the sample mean and sample variance, or do you observe two numbers which you are modeling as single samples from two independent normal random variables with known means and variances? Note that @Max's comment implicitly assumes independence of the two samples. – Dilip Sarwate Oct 27 '12 at 03:22
  • Without additional details on the first part this is unanswerable as it's impossible to tell which of a host of possible things is being asked for. – Glen_b Jun 21 '16 at 12:58
  • I think this question is correct. this is simple way of merging two distribution-http://pballew.net/combdis.htm – Arpit Sisodia Feb 24 '17 at 04:30
  • @ Max I believe your answer is correct except that it is not the curves that are averaged just the mean for the first question. For the second, the applicable mixture distribution is just $\frac{1}{2}N(\mu_1,\sigma_1)+\frac{1}{2}N(\mu_2,\sigma_2)$. – Carl Feb 24 '17 at 04:51

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