1. Analysis of RT distributions with R Emil Ratko-Dehnert WS 2010/ 2011 Session 03 – 23.11.2010

2. 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

3. RANDOM VARIABLES & THEIR CHARACTERIZATION 3 II

4. Random Variables Usually one is not interested in the probabili- ties of single events ω from Ω Rather one wants to know specific features of the whole space 4 II

5. Random variables (2) „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) 5 II

6. Examples 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 II 6

7. 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) II 7

8. RV distribution 8 0 1 Ω P A 0 R X For an event A = [a, b[, a, b in R: P X (A) = P(X -1 (A)) II x PXPX

9. In our case... 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 P X from now on 9 II

10. Characterization of RVs 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,... 10 II

11. Mean(X) = X = E(X) What is the expected (long term) outcome of X? Mean(X), X, μ or Expected Value E(X) Discrete Continuous 11 II

12. Example: Unfair.dice 12 Weighted sum II

13. Characterizing Unfair.dice 13 X = 4.25 II

14. Variance „How much do the values of a RV X vary around its mean value X ?“ Discrete Continuous 14 II

15. What is the standard deviation? „The standard deviation sd(X) or σ X is the square root of the variance of X.“ 15 II

16. Characterizing Unfair.dice 16 X = 4.25 II var(X) = 2.55 σ X = 1.6 σXσX σXσX

17. Mode(X) and Median(X) Mode(X) = value with highest probability Median X med = value, splitting upper from lower half of values (w. r. t. P) 17

18. Characterizing Unfair.dice 18 X = 4.25 II med(X) = 5 mod(X) = 6

19. AND NOW TO 19