Session-04
Discovering the Binomial Distribution from First Principles
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Why the Bell Curve Shows Up Everywhere in Data and Nature
Each of the 100 users is independently routed to server 1 with probability $1/5 = 0.2$. This sounds exactly like a coin flip repeated $n = 100$ times with $p = 0.2$. What distribution does the count follow? What is its expected value?
$X \sim \text{Binomial}(100, 0.2)$ with $E[X] = np = 20$. This is the canonical model for counting successes in independent trials. The next segment derives the Binomial PMF from first principles, building up to the general formula $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$.
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