Multiplying by 0 ( zero ) when finding max-likelihood estimation

Question

I am kind of confused about the rules in math when you are solving equations I think.

I am solving the following equation: $\frac{15}\lambda - \sum_{i=1}^{15} X_i = 0 $

According to maple and symbolab, in order to solve this equation, you are to start with multiplying by lambda on each side.

This would give us $(\frac{15}\lambda - \sum_{i=1}^{15} X_i)\lambda = 0 * \lambda $, then $15 - \lambda\sum_{i=1}^{15} X_i = 0$, then $\lambda\sum_{i=1}^{15} X_i = 15$, then $\lambda = \frac{15}{\sum_{i=1}^{15} X_i}$

This is also the right answer according to my professor.

Instead, I get $\lambda = \frac{\sum_{i=1}^{15} X_i}{15}$

by $\frac{15}\lambda - \sum_{i=1}^{15} X_i = 0 $, then $\frac{15}\lambda = \sum_{i=1}^{15} X_i$, then $\lambda = \frac{1}{15} * \sum_{i=1}^{15} X_i$

This is however a completely different answer and it is wrong, even though I think I'm not doing anything illegal (I think? Please correct me if I'm wrong)

Is there anyone here that can maybe spot my mistake and maybe tell me why we are allowed to multiply by 0 ( zero ) on one of the sides here? Sorry if the math format is a bit bad, could not find left right arrow.

• Zebraboard

Answer 1

in your latest passage you have

$$\frac{15}{\lambda}=\Sigma_iX_i$$

that is

$$\frac{1}{\lambda}=\frac{\Sigma_iX_i}{15}$$

taking the reciprocal of both memebers you get the same solution as your professor.

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Original equation

$$\frac{15}{\lambda}-\Sigma_iX_i=0$$

1. sum $\Sigma_iX_i$ to both members getting

$$\frac{15}{\lambda}=\Sigma_iX_i$$

2. divide both member by $15$

$$\frac{1}{\lambda}=\frac{\Sigma_iX_i}{15}$$

3. take the reciprocal of both members getting the solution