Add more changes to dig and prob

1 files changed, 20 insertions, 20 deletions

diff --git a/sem6/prob/m3/noter.md b/sem6/prob/m3/noter.md
index 23990ef..c784a6a 100644
--- a/sem6/prob/m3/noter.md
+++ b/sem6/prob/m3/noter.md
@@ -19,7 +19,7 @@ $$
     E[X] = \int_{\infty}^{\infty} x f(x) \mathrm{dx}
 $$
 
-Can also calculate expectation distribution function:
+Can also calculate expectation distribution function, however this can only be used if all values are non-negative:
 
 $$
     E[X] = \sum_{k=0}^{\infty} P(X > k) \\
@@ -55,7 +55,7 @@ $$
     E[Z] = \sum_{i} \sum_{j} g(x_i, y_j) \cdot p(x_i, y_j)
 $$
 
-If discrete just use integrals instead.
+If continues just use integrals instead.
 
 $$
     E[Z] = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(x, y) \cdot f(x, y) \mathrm{dxdy}
@@ -69,27 +69,27 @@ $$
 
 If $X$ and $Y$ are **independent** the following is true:
 
-$$
-    E[g_1(X) \cdot g_2(Y)] = E[g_1(X)] \cdot E[g_2(Y)] \\
-    E[X \cdot Y] = E[X] \cdot E[Y]
-$$
+\begin{align*}
+    E[g_1(X) \cdot g_2(Y)] &= E[g_1(X)] \cdot E[g_2(Y)] \\
+    E[X \cdot Y] &= E[X] \cdot E[Y]
+\end{align*}
 
 ## Variance
 
 Describes the mean of the distance between outcomes and the overall mean.
 Good way to describe the spread of the random variable.
 
-$$
-Var(X) = E[(X - E[X])^2] \\
-Var(X) = E[X^2] - E[X]^2
-$$
+\begin{align*}
+    Var(X) &= E[(X - E[X])^2] \\
+    Var(X) &= E[X^2] - E[X]^2
+\end{align*}
 
 If there is no power of two, it will be mean minus mean, which wont work.
 
 One can define the *standard deviation* to bring back the unit from squared.
 
 $$
-    Std(X) = \sqrt{ (Var(X)) }
+    Std(X) = \sqrt{ Var(X) }
 $$
 
 A rule for variance:
@@ -110,21 +110,21 @@ If X and Y are independent the Cov part disappears.
 
 ## Covariance
 
-$$
-    Cov(X,Y) = E[(X - E[X]) \cdot (Y - E[Y])] \\
-    Cov(X,Y) = E[XY] - E[X] \cdot E[Y]
-$$
+\begin{align*}
+    Cov(X,Y) &= E[(X - E[X]) \cdot (Y - E[Y])] \\
+    Cov(X,Y) &= E[XY] - E[X] \cdot E[Y]
+\end{align*}
 
 Shows whether two variables vary together, can be both positive and negative.
 If it is possible $X$ and $Y$ are varying from the average together.
 
 Some rules below:
 
-$$
-    Cov(X, X) = Var(X) \\
-    Cov(a X, Y) = a Cov(X, Y) \\
-    Cov(X + Y, Z) = Voc(X, Z) + Cov(Y, Z)
-$$
+\begin{align*}
+    Cov(X, X) &= Var(X) \\
+    Cov(a X, Y) &= a Cov(X, Y) \\
+    Cov(X + Y, Z) &= Cov(X, Z) + Cov(Y, Z)
+\end{align*}
 
 If X and Y are independent, then covariance is zero (X and Y are *uncorrelated*).
 X and Y can be uncorrelated and not be independent.