I feel very silly asking this question but I am having some trouble with the basic rules of product operators and it is causing trouble in my calculation of likelihood functions. I will share the specific problem below:
$$L(\theta) = \prod (\theta+1) y^\theta$$
When I attempt to simplify the above, I begin by writing out the equation:
$$= (\theta + 1) y_1^\theta \cdot (\theta + 1) y_2^\theta \cdots (\theta + 1) y_n^\theta$$
Then, my impulse is to factor out $(\theta + 1)$, since it is a constant, and be left with:
\begin{align} & = (\theta + 1)(y_1^\theta \cdot y_2^\theta \cdots y_n^\theta) \\[10pt] & = (\theta + 1) \prod y_i^\theta \end{align}
This is incorrect, and I am supposed to obtain:
$$(\theta + 1)^n \prod y_1^\theta$$
I guess I am confused about why I can't "factor out" the $(\theta + 1)$, and why it is ultimately raised to the nth power.
Any advice would be great! I feel silly asking what is essentially algebra but I have never worked with product operations before and I guess am having some issues.
You have $$(\theta+1)\cdot y_1^\theta\cdot (\theta+1)\cdot y_2^\theta\cdots(\theta+1)\cdot y_n^\theta$$ i.e. $n$ lots of $(\theta+1)y_i^\theta$ all multiplied together.
Then since you can re-order multiplications, you can take all the $(\theta+1)$'s to the front. There are $n$ of these. So you get $$\underbrace{(\theta+1)\times(\theta+1)\times...\times(\theta+1)}_{n\text{ times}}\cdot y_1^\theta\cdots y_n^\theta=(\theta+1)^n\cdot y_1^\theta\cdots y_n^\theta$$
When you "factor out", this is usually done from a sum - you take common factors out of multiple terms which are added together. E.g: $$2x+3x+4x=(2+3+4)x$$ Here you don't have a sum, you have a product, so there is only 1 term (in terms of addition). As an analogue to the above example, its like having $$2x\times 3x\times 4x=x^3(2\times 3\times 4)$$