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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) 5 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 6 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) 7 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)) 8 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 9 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, ... 10 II Mean(X) = X = E(X) • What is the expected (long term) outcome of X? • Mean(X), X, μ or Expected Value E(X) • Discrete E ( X ) : x P( x ) i i i • Continuous E ( X ) X P( X ) dx 11 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 12 Characterizing Unfair.dice X = 4.25 II 13 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 14 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 ) 15 Characterizing Unfair.dice var(X) = 2.55 σX = 1.6 σX X = 4.25 II σX 16 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) 17 Characterizing Unfair.dice II mod(X) = 6 X = 4.25 med(X) = 5 18 AND NOW TO 19