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Home » Length: \( 3w = 150 \) meters - Silent Sales Machine

Length: \( 3w = 150 \) meters - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 For $k = 0,1,2,3$: output = $y = 8 - k$

📌 For $k = 4,5,6,7,8$: output = $k$

📌 \mathbb{E}[\text{output}] = \sum_{k=0}^{3} (8 - k) \cdot \frac{\binom{8}{k}}{256} + \sum_{k=4}^{8} k \cdot \frac{\binom{8}{k}}{256}

📌 Compute each sum.

📌 First sum ($k = 0$ to $3$):

📌 $ (8-0)\binom{8}{0} = 8 \cdot 1 = 8 $

📌 $ (8-1)\binom{8}{1} = 7 \cdot 8 = 56 $

📌 $ (8-2)\binom{8}{2} = 6 \cdot 28 = 168 $

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If \( x = -3 \), then \( x - 1 = -4 < 0 \), invalid.
If \( x = 3 \), then \( x - 1 = 2 > 0 \), valid.
Thus, the only solution is \( x = 3 \).
A quadratic function passes through the points (1, 4), (2, 3), and (4,...
Let the quadratic be \( f(x) = ax^2 + bx + c \).
Using the points:
For (1,4): \( a + b + c = 4 \)
For (2,3): \( 4a + 2b + c = 3 \)
For (4,15): \( 16a + 4b + c = 15 \)
Subtract first from second: \( (4a + 2b + c) - (a + b + c) = 3 - 4 \Ri...

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