how come this summation after produkt?

Question

I am really stuck in this step. I hope, the context does not matter here, so i didnot provide what this is about. I am trying to get ML-Estimator. but the problem is, as i see in my textbook, how they changed the produkt to summation, why it became $\sum$

$$L(a;X)=\prod_{i=1}^{n}\frac{1}{2}a \cdot exp(-a\cdot|x_i|) = \frac{1}{2^n}a^n\cdot exp(-a\sum_{i=1}^{n}|x_i|)$$

THANKS for help!

$\exp(a)\exp(b)=\exp(a+b)$ – André Nicolas Jun 28 '13 at 07:11 — André Nicolas, Jun 28 '13 at 07:11

Did · Answer 1 · 2013-06-28T07:39:17.297

2

$$\prod_{i=1}^n\left(a\cdot\mathrm e^{u_i}\right)=a^n\cdot\prod_{i=1}^n\mathrm e^{u_i}=a^n\cdot\exp\left(\sum_{i=1}^nu_i\right)$$

edited Jun 28 '13 at 07:39

answered Jun 28 '13 at 07:12

Did

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This is the great disadvantage of adding no explanation about the problem you have with the question: answerers cannot know what is troubling you. Perhaps André's comment is all you need, how are we supposed to know? See Edit. – Did Jun 28 '13 at 07:26
((Comment by the OP cancelled after I answered it.)) – Did Jul 07 '13 at 16:16

score 1 · Accepted Answer · answered Jun 28 '13 at 07:11

1

This is just a property of the exponential function. Indeed $\exp(a+b)=\exp(a)\cdot\exp(b)$ and so induction shows that $$ \prod_{i=1}^n\exp(a_i)=\exp\left(\sum_{i=1}^n a_i\right). $$ Use this with $a_i=-\vartheta|x_i|$.

answered Jun 28 '13 at 07:11

Stefan Hansen

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oh yeah, this is the point, thanks Stefan – doniyor Jun 28 '13 at 07:23

how come this summation after produkt?

2 Answers2