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+# Notes for third probability lecture
+
+*Moment generation function* is a third way to describe the behavior of a random variable.
+
+Can also find *Key performance indicators*.
+Which will be explained in this document.
+
+## Expectation
+
+Is just the mean value.
+
+$$
+    E[X] = \sum_{i} x_i P(X = x_i) = \sum_{i} x_i p(x_i)
+$$
+
+In the continues way integral is used instead.
+
+$$
+    E[X] = \int_{\infty}^{\infty} x f(x) \mathrm{dx}
+$$
+
+Can also calculate expectation distribution function:
+
+$$
+    E[X] = \sum_{k=0}^{\infty} P(X > k) \\
+    E[X] = \int_{0}^{\infty} (1 - F(x)) \mathrm{dx}
+$$
+
+Tatianas recommendation is to calculate expectation from the PDF.
+
+
+
+
+### LOTUS
+
+
+### Some properties
+
+$$
+E[a X + b] = a E[X] + b
+$$
+
+The mean of a constant is the constant itself:
+
+$$
+E[b] = b
+$$
+
+
+### Multiple variables
+
+If $Z = g(X,Y)$ one can find the expectation with:
+
+$$
+    E[Z] = \sum_{i} \sum_{j} g(x_i, y_j) \cdot p(x_i, y_j)
+$$
+
+If discrete just use integrals instead.
+
+$$
+    E[Z] = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(x, y) \cdot f(x, y) \mathrm{dxdy}
+$$
+
+The following rule can be used:
+
+$$
+E[X+Y] = E[X] + E[Y]
+$$
+
+If $X$ and $Y$ are **independent** the following is true:
+
+$$
+    E[g_1(X) \cdot g_2(Y)] = E[g_1(X)] \cdot E[g_2(Y)] \\
+    E[X \cdot Y] = E[X] \cdot E[Y]
+$$
+
+## Variance
+
+Describes the mean of the distance between outcomes and the overall mean.
+Good way to describe the spread of the random variable.
+
+$$
+Var(X) = E[(X - E[X])^2] \\
+Var(X) = E[X^2] - E[X]^2
+$$
+
+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.
+
+$$
+    Std(X) = \sqrt{ (Var(X)) }
+$$
+
+A rule for variance:
+
+$$
+Var(a X + b) = a^2 Var(X)
+$$
+
+The variance of a constant is therefore $0$.
+
+### Summing
+
+$$
+Var(X+Y) = Var(X) + Var(Y) + 2 Cov(X, Y)
+$$
+
+If X and Y are independent the Cov part disappears.
+
+## Covariance
+
+$$
+    Cov(X,Y) = E[(X - E[X]) \cdot (Y - E[Y])] \\
+    Cov(X,Y) = E[XY] - E[X] \cdot E[Y]
+$$
+
+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:
+
+$$
+    Cov(X, X) = Var(X) \\
+    Cov(a X, Y) = a Cov(X, Y) \\
+    Cov(X + Y, Z) = Voc(X, Z) + Cov(Y, Z)
+$$
+
+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.
+
+## Correlation coefficient
+
+It is hard to compare covariance as the value is dependent on the size of X and Y values.
+We can therefore take the Correlation coefficient instead.
+
+$$
+    Corr(X,Y) = \frac {Cov(X,Y)} {\sqrt{Var(X)} \cdot \sqrt{Var(Y)}}
+$$
+
+    vim: spell spelllang=da,en