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Diffstat (limited to 'sem6/prob/m3/noter.md')
| -rw-r--r-- | sem6/prob/m3/noter.md | 40 |
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. |