## ~shreyasminocha/uni-notes

b343c48e746986e5a0537bc5a748314533f92655 — Shreyas Minocha a month ago
Update

2 files changed, 28 insertions(+), 7 deletions(-)

M comp/182.md
M elec/220.md

M comp/182.md => comp/182.md +15 -7
@@ 767,7 767,7 @@ $$\left(x + y\right)^n = \sum_{i = 0}^n \binom{n}{i} x^{n - i} y^i$$

**pascal's identity**: $\binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}$

-**vandermonde's identity**: $\binom{m + n}{k} = \sum{l = 0}{k} \binom{m}{k - l} \binom{n}{l}$
+**vandermonde's identity**: $\binom{m + n}{k} = \sum_{l = 0}^{k} \binom{m}{k - l} \binom{n}{l}$

## Disscrete Probability

@@ 869,24 869,32 @@ Run $n$ independent Bernoulli trials and count the number of successes in the se

$p(X = k) = \binom{n}{k} p^k {(1 - p)}^{n - k}$

-$\mathbb{E}(x) = np$
+$\mathbb{E}(X) = np$

-$V(x) = np(1 - p)$
+$V(X) = np(1 - p)$

#### Geometric Distribution

-$p(x) = {(1 - p)}^{k - 1} p$
+$p(X = k) = {(1 - p)}^{k - 1} p$

-$\mathbb{E}(x) = \frac{1}{p}$
+$\mathbb{E}(X) = \frac{1}{p}$

-$V(x) = \frac{1 - p}{p^2}$
+$V(X) = \frac{1 - p}{p^2}$

#### Negative Binomial Distribution

+$l$ trials until we see $k$ successes
+
+$p(X = l) = \binom{l - 1}{k - 1} {(1 - p)}^{l - k} p^k$
+
#### Poisson Distribution

### Markov Chains and Hidden Markov Models (HMMs)

+-   initial distribution ($\pi$)
+-   emission probability ($E_l(X_i)$)
+-   transition matrix ($A_{l', l}$)
+
### Parameter Estimation For Markov Chains

## Dynamic Programming

@@ 973,7 981,7 @@ If there exists a 1-1 function $f : X \rightarrow Y$, then $|X| \leq |Y|$.

There exists a bijection $f : X \rightarrow Y$ iff there exists a bijection $g : Y \rightarrow X$.

-**Shröder-Bernstein Theorem**: if $A$ and $B$ are two sets such that $|A| \leq |B|$ and $|B| \leq |A|$, then $|A| = |B|$.
+**Shröder-Bernstein Theorem**: if $A$ and $B$ are two sets such that $|A| \leq |B|$ and $|B| \leq |A|$, then $|A| = |B|$. Show a one-to-one function $f : A \rightarrow B$ and a one-to-one function $g : B \rightarrow A$.

### An infinity of infinities


M elec/220.md => elec/220.md +13 -0
@@ 302,3 302,16 @@ locality:

-   temporal locality
-   spatial locality
+
+types:
+
+-   direct-mapped cache
+-   set associative cache
+
+$C = S \times E \times B$ data bytes
+
+-   $S = 2^s$
+-   $E = 2^e$. when direct-mapped, $e = 1$
+-   $B = 2^b$
+
+Word length: $t + s + b$