Equation 14 · Part 9 · When the Substrate Became the Product

Starting index or lower bound: i=1

Y_{\mathrm{pkg}} = Y_{\mathrm{assy}} \prod_{i=1}^{N} Y_i .

i=1

What this part means

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

Its job in the formula

i=1 appears in the bound of this product. The bound states where the repeated operation starts, ends, or which values it includes.

Full expressionStarting index or lower bound: i=1Article meaning

The passage around this formula

A multi-die package inherits a property that a single die does not have: its yield is a product, not a term. If a package contains dies with individual probabilities $Y_i$ of being good, and the assembly itself succeeds with probability $Y_{\mathrm{assy}}$ , then $Y_{\mathrm{pkg}} = Y_{\mathrm{assy}} \prod_{i=1}^{N} Y_i$ . Multiplication is brutal at scale. Eight components at 99 percent each land near 92 percent together before assembly loss is counted; the same eight at 95 percent land near 66 percent. And the loss is not proportional to the failing part — scrapping a package destroys every good die in it plus the assembly work. This is precisely why known-good-die testing dominates the discussion: the whole point is to move the $Y_i$ …

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Learn the underlying idea

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

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Sources cited in the article section

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