Solving equations in SageMath

Hey! I'm new to sage and I'm trying to see how complex are the calculations that I can express in SageMath. So I considered the following example, the comparison between the likelihood of two normals with different means but same variance:

$$\dfrac{1}{\sigma\sqrt{2\pi} } \exp{\left(-\dfrac{(x-\mu_1)^2}{2\sigma^2}\right)} = \dfrac{1}{\sigma\sqrt{2\pi} } \exp{\left(-\dfrac{(x-\mu_2)^2}{2\sigma^2}\right)}$$

I know that the answer must be $\mu_1 = \mu_2$. Can I solve it with SageMath? I tried:

x, mu1, mu2, sigma = var('x, mu1, mu2, sigma')
exp((0.5/sigma)*(x-mu1)^2)==exp((0.5/sigma)*(x-mu2)^2)


and

solve(exp((0.5/sigma)*(x-mu1)^2)==exp((0.5/sigma)*(x-mu2)^2))


but that doesn't seem to work. Is it possible to do so in sagemath? Thanks!!

edit retag close merge delete

Sort by ยป oldest newest most voted

You can solve for mu1:

var('x, mu1, mu2, sigma')
result=solve(exp((0.5/sigma)*(x-mu1)^2)==exp((0.5/sigma)*(x-mu2)^2),mu1)
show(result)

more

Thanks, that worked!!

( 2021-07-20 15:39:21 +0200 )edit

By the way, what if I wanted to solve for two variables, meaning I wanted to check if for different $\sigma_1$ and $\sigma_2$, I tried the following:

var('x, mu1, mu2, sigma1, sigma2')
result = solve((1/sigma1^2)*(x-mu1)^2 + log(sigma1)==(1/sigma2^2)*(x-mu2)^2+log(sigma2), mu1, sigma1)


But it didnt't work out. How can I define for several variables the solution?

( 2021-07-20 15:56:36 +0200 )edit
1

Usually, you need a second equation to solve for two variables.

( 2021-07-20 21:15:32 +0200 )edit