How $\prod_{j=k+1}^{n}(e^{-\lambda t_k})=e^{-(n-k)\lambda t_k}$?

Question

I was solving one MLE(Maximum Likelihood Estimator) question and I encountered for this equation. How could I simplify $\prod_{j=k+1}^{n}(e^{-\lambda t_k})$ to the given answer? Because I'm unfamiliar with the $\prod$. Hope someone could help me out.

$$\prod_{j=k+1}^{n}(e^{-\lambda t_k})=e^{-(n-k)\lambda t_k}$$

Answer 1

$$\begin{align} \prod_{j=k+1}^ne^{-\lambda{t_k}} &=\overbrace{e^{-\lambda{t_k}}\cdot e^{-\lambda{t_k}}\cdot e^{-\lambda{t_k}}\cdots e^{-\lambda{t_k}}}^{n-(k+1)+1=n-k\text{ times}}\\ &=e^{-(\overbrace{\lambda t_k+\lambda t_k+\lambda t_k+\cdots+\lambda t_k}^{n-k\text{ times}})}\\ &=e^{-(n-k)\lambda t_k}\end{align}$$

Answer 2

The $\prod_{j=k+1}^{n}$ in

$$\prod_{j=k+1}^{n}(e^{-\lambda t_k})=e^{-(n-k)\lambda t_k}$$

can be read as taking $n - k$ copies of the quantity after it, and multiplying it all together. Consider, for example, the case where $j = k + 1 = n$. Then it will simply be a single "copy". For $j = k + 1 = n - 1$, the "loop" will run for two iterations, etc.

Taking $n - k$ "copies" of that quantity behind it is equivalent to simply raising it to the power $(n - k)$. Since there already is a power up there, if you raise it to another power, you can simply multiply the powers. Doing that directly gives you the result.

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Note that the above is only correct in your specific case because the "index" $j$ (the symbol at the bottom of the $\prod$) does not appear in the expression after the product symbol. If the $j$ would appear in that following expression, you would not multiply a sequence of "copies", but a sequence of expressions where you fill in varying $j$, starting with $j = k + 1$ and ending with $j = n$.

For example:

$$\prod_{j=1}^3 j = 1 \times 2 \times 3$$

or

$$\prod_{j=k+1}^n e^j = e^{k+1} \times e^{k+2} \times \dots \times e^{n}$$

Answer 3

Starting from $k+1$, how many steps are required to reach $n$? $k+1, k+2, ...,n.$ $$k+1(steps) = n$$ $$steps = n-k$$

As you aren't familiar with the $\prod$ its just a notation for multiplication of the terms. For e.g:

$$\prod_{j=1}^{3}x^j=x^1\cdot x^2\cdot x^3$$ Moreover, $$2^3 = 2\cdot 2\cdot 2 = \prod_{j=1}^{3}2$$ the term in the product is just repeated a number of times when it does not depend on the index. Hence, Let $C = e^{-\lambda t_k}$ $$\prod_{j=k+1}^{n}(C)=C^{n-k}$$ $$C = e^{-\lambda t_k}$$ $$C^{n-k} = ({e^{-\lambda t_k}})^{n-k} = e^{-(n-k)\lambda t_k}$$