~shreyasminocha/uni-notes

c29d619b2eb67f1e0010ccb2ce6c12ad182d5df4 — Shreyas Minocha a month ago cb93232
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@@ 825,10 825,20 @@ $\mathbb{E}(X) = \sum_{s \in S} p(s) \cdot X(s) = \sum_x p(x) \cdot x$

If $X_1, X_2, \ldots, X_n$ are rv's on $S$, then $\mathbb{E}(X_1 + X_2 + \ldots + X_n) = \mathbb{E}(X_1) + \mathbb{E}(X_2) + \cdots + \mathbb{E}(X_n)$

$\mathbb{E}(aX + b) = a\mathbb{E}(X) + b$

For independent random variables $X$ and $Y$, $\mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y)$.

### Average case running time of linear search

### Variance

$var(X) = \sum_x \left(x - \mathbb{E}\right)^2 - p(x)$

$var(X) = \mathbb{E}[\left(X - \mathbb{E}\right)^2]$

$var(X) = \mathbb{E}(X^2) - \left[\mathbb{E}(X)\right]^2$

## Discrete Probability

### Tail Bounds