Samenvatting Advanced Methods for Applied Spatial Economic Research 2013-2014
Background (SW 2&3)
- outcomes: mutually exclusive (only one will actually occur) potential results of a random process
- probability of an outcome: proportion of the time that a specific outcome will occur in the long run.
- sample space: set of all possible outcomes
- event: a set consisting of two or more outcomes
- random variable: a numerical summary of a random outcome (discrete: ‘whole’ values like 1,2 or 3
and continuous)
- probability distribution: a list of all possible values of a variable and the probability that each value
will occur, the probabilities sum to 1
- Bernoulli random variable: binary discrete random variable (outcome of 0 or 1)
- Bernoulli distribution: outcome 0 has a probability of 1-p, outcome 1 of p
- expected value E(Y): long run average value of a random variable Y, for discrete random variables it
is a weighted average (∑possible values x their probability).
- mean: the expected value, denoted as: µY
- expected value of Bernoulli variable:1 x p + 0 x (p-1) = p
- sample mean: the mean value of n samples is, = (Y1 + Y2 + … Yn)
- variance, denoted as var(Y) or бY 2 : measurement of the spread of a probability distribution. Given
by: ∑(yi - µY )2 pi
- standard deviation, denoted as б Y :
- variance and standard deviation of Bernoulli: б Y 2 = p(1-p) and бY =
- skewness: how much a distribution deviates from symmetry, [∑(yi - µY )3 pi ]/ бY 3
- kurtosis: how much mass is in the tails. The greater the kurtosis, the more likely are outliers. [∑(yi -
µY )4 pi ]/ бY 4
- covariance, denoted as cov(X,Y) or б XY : the extent to which two random variables move together,
given by: ∑(xj - µX)(yi - µY )P(xj, yi )
- correlation, denoted as corr(X,Y): the dependence between X and Y, very similar to the covariance.
The formula is: corr(X,Y) = cov(X,Y)/ = бXY / бX бY
* Distributions
- Normal: bell shaped probability density. The mean µ lies in the center. This distribution has 95% of
its probability between µ - 1,96б and µ + 1,96б.
- Chi-squared: sum of m squared independent standard normal random variables. The degrees of
freedom are equal to the number m of used variables.
-Student t: m degrees of freedom. The distribution of the ratio of a standard normal random variable
divided by the square root of an independently distributed chi-squared random variable divided by m.
For example: Z/
- F distribution: W is a chi-squared random variable with m degrees of freedom. V is a chi-squared
random variable with n degrees of freedom. The variables are independently distributed. In formula:
(W/m)/(V/n)
- bar: mean
- dakje: estimator
- tilde: relatie/verandering
Background (SW 2&3)
- outcomes: mutually exclusive (only one will actually occur) potential results of a random process
- probability of an outcome: proportion of the time that a specific outcome will occur in the long run.
- sample space: set of all possible outcomes
- event: a set consisting of two or more outcomes
- random variable: a numerical summary of a random outcome (discrete: ‘whole’ values like 1,2 or 3
and continuous)
- probability distribution: a list of all possible values of a variable and the probability that each value
will occur, the probabilities sum to 1
- Bernoulli random variable: binary discrete random variable (outcome of 0 or 1)
- Bernoulli distribution: outcome 0 has a probability of 1-p, outcome 1 of p
- expected value E(Y): long run average value of a random variable Y, for discrete random variables it
is a weighted average (∑possible values x their probability).
- mean: the expected value, denoted as: µY
- expected value of Bernoulli variable:1 x p + 0 x (p-1) = p
- sample mean: the mean value of n samples is, = (Y1 + Y2 + … Yn)
- variance, denoted as var(Y) or бY 2 : measurement of the spread of a probability distribution. Given
by: ∑(yi - µY )2 pi
- standard deviation, denoted as б Y :
- variance and standard deviation of Bernoulli: б Y 2 = p(1-p) and бY =
- skewness: how much a distribution deviates from symmetry, [∑(yi - µY )3 pi ]/ бY 3
- kurtosis: how much mass is in the tails. The greater the kurtosis, the more likely are outliers. [∑(yi -
µY )4 pi ]/ бY 4
- covariance, denoted as cov(X,Y) or б XY : the extent to which two random variables move together,
given by: ∑(xj - µX)(yi - µY )P(xj, yi )
- correlation, denoted as corr(X,Y): the dependence between X and Y, very similar to the covariance.
The formula is: corr(X,Y) = cov(X,Y)/ = бXY / бX бY
* Distributions
- Normal: bell shaped probability density. The mean µ lies in the center. This distribution has 95% of
its probability between µ - 1,96б and µ + 1,96б.
- Chi-squared: sum of m squared independent standard normal random variables. The degrees of
freedom are equal to the number m of used variables.
-Student t: m degrees of freedom. The distribution of the ratio of a standard normal random variable
divided by the square root of an independently distributed chi-squared random variable divided by m.
For example: Z/
- F distribution: W is a chi-squared random variable with m degrees of freedom. V is a chi-squared
random variable with n degrees of freedom. The variables are independently distributed. In formula:
(W/m)/(V/n)
- bar: mean
- dakje: estimator
- tilde: relatie/verandering
