Transcript Analysis of RT distributions with R - uni

Analysis of RT distributions
with R
Emil Ratko-Dehnert
WS 2010/ 2011
Session 03 – 23.11.2010
Last time ...
• Introducing of Probability Space (Ω, A, P) with
examples
• Kolmogorov axioms (-> contraints for modelling)
• Discrete vs continuous distributions
• Law of large numbers (-> aggregation to the mean)
• Central limit theorem (-> normality of errors)
• Matrix Calculus (-> mind dimensions and operations)
2
II
RANDOM VARIABLES &
THEIR CHARACTERIZATION
3
Random Variables
II
• Usually one is not interested in the probabilities of single events ω from Ω
• Rather one wants to know specific features of
the whole space
4
Random variables (2)
II
• „A random variable (RV) X, is a variable whose
outcomes are probabilistic“
• Formal definition:
A random variable X is a (measurable) mapping
from the probability space to the reals:
X: Ω  R;
ω  X(ω)
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Examples
II
• Rolling two dice, Ω = { ω = (i,j) }
 X(i,j) = i + j
• Betting on heads/ tails, Ω = {„head“, „tail“}
X(„head“) = 20 and X („tail“) = - 5
 gain/ cost function
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II
RV calculus
• ( X + Y ) (ω) = X(ω) + Y(ω)
(additivity)
• ( a * X ) (ω) = a * X(ω)
(scalar multipl.)
• ( X * Y ) (ω) = X(ω) * Y(ω)
(multipl.)
• Likewise: min(X), max(X) , f(X)
(functions)
(for f Borel-measurable)
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RV distribution
II
Ω
P
A
1
0
PX
X
0
x
R
For an event A = [a, b[, a, b in R: PX (A) = P(X-1(A))
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In our case...
II
• Reaction times of behavioural experiments are RV‘s
• Fortunately here, matters are less complicated:
Ω = R; X = id (!)
• This means, we can simply investigate the original
probability distribution P instead of PX from now on
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Characterization of RVs
II
• How can RV distributions be (reasonably)
characterized?
• By the moments of its distribution: mean,
variance, (curtosis, skewness, ...)
• By descriptive statistics: mode, median, quantiles
• By its distributional parameters: μ, σ, λ, ...
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Mean(X) = X = E(X)
II
• What is the expected (long term) outcome of X?
• Mean(X), X, μ or Expected Value E(X)
• Discrete
E ( X ) :  xi  P( xi )
i
• Continuous
E ( X )   X  P ( X ) dx
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Example: Unfair.dice
xi  1, 2, 3, 4, 5, 6
II
1 1 1 1 1 1 
P( xi )   , , , , , , 
 12 12 6 6 6 3 
Weighted sum
1
1
1
1
1
1
X  1
 2
 3   4   5   6   4.25
12
12
6
6
6
3
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Characterizing Unfair.dice
X = 4.25
II
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Variance
II
• „How much do the values of a RV X vary around
its mean value X ?“
• Discrete
Var ( X )  E( X  X )   pi  ( xi  X )
2
2
i
• Continuous
Var ( X )   ( x   )  p( x)dx
2
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What is the standard deviation?
II
• „The standard deviation sd(X) or σX is the
square root of the variance of X.“
 X  Var(X )
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Characterizing Unfair.dice
var(X) = 2.55
σX = 1.6
σX
X = 4.25
II
σX
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Mode(X) and Median(X)
• Mode(X) = value with highest probability
• Median Xmed = value, splitting upper from
lower half of values (w. r. t. P)
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Characterizing Unfair.dice
II
mod(X) = 6
X = 4.25
med(X) = 5
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AND NOW TO
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