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--- math121b:04-15 [2020/04/15 05:37]
pzhou created
+++ math121b:04-15 [2026/02/21 14:41] (current)
@@ Line 15: / Line 15: @@
 ** Conditional Probability **: suppose we want to know "given that $A$ happens, what is the probability $B$ will happen?"
-$$ \P(A | B) := \frac{ \P(A \cap B) } {\P(B) \} $$
+$$ \P(A | B) := \frac{ \P(A \cap B) } {\P(B) } $$
 ** The product rule **
@@ Line 45: / Line 45: @@
 This is a function of $x$, we just need to 'renormalize' it so that the integral of $x$ over $\R$ is $1$. This gives
 $$  \rho_{X|Y}(x|y=y_0) = \frac{ \rho_{XY}(x, y_0)}{ \int_\R \rho_{XY}(x', y_0) dx' }. $$
+===== How to play the probability game? =====
+First thing first, find out what is $\Omega$ and $\P$.
+Be aware when someone say: "let me randomly choose ...."
+Here is an interesting example: [[https://en.wikipedia.org/wiki/Bertrand_paradox_(probability) | Bertrand Paradox ]]
+Another hard example: let $M$ be a random symmetric matrix of size $N \times N$, where each entry is iid Gausian. Question: how does eigenvalues of this matrix distribute?

Lecture Notes