Probability

 Probability values range from 0 to 1.  Adding all probabilities of the sample yields 1.  The probability that an event A will not occur is 1 minus the probability of A.  If two events are independent, the probability that one or the other event occurs is the sum of their individual probabilities. Principles of probability calculations

Simple probability P(A) = 1/6 = Sample space:1,2,3,4,5,6

Joint probability P(5,6) = P(A,B) = P(A)  P(B) P(0.166)  P(0.166) =

Joint probability -> V NP PP -> V [NP PP] (1)keep the dogs on the beach

keep VP VP → V NP XP [.15] V NP PP the dogson the beach keep: V NP XP [.81].15 x.81 =.12 Conditional probability

keep VP VP → V NP XP [.15] V NPPP the dogson the beach keep: V NP [.19].19 x.39 x 14 =.01 NP NP → NP PP [.14] Conditional probability

In a corpus including nouns and adjectives, adjectives precede a noun. What is the likelihood that a noun occurs after an adjective? P(2000) P(12000) P(ADJ|N)=

Conditional probability What is the likelihood that an adjective precedes a noun? P(2000) P(3500) P(N|ADJ)=

Probability distribution

 Discrete probability distribution  Continuous probability distribution Types of probability distributions

Binomial distribution

 two possible outcomes on each trail  the outcomes are independent of each other  the probability ratio is constant across trails Bernoulli trail: Binomial distribution

T H HHHTTHTT Binomial distribution

0 heads= HH 1 head=HT + TH 2 heads=TT Binomial distribution

HH HT TH TT Sample spaceRandom variable Binomial distribution

Cumulative outcomeProbability 0 = 1  1 = 2  2 = 1   P(x) = 1 Binomial distribution

T H HH HTTHTT HHHHHTHTHHTTTHHTHTTTHTTT

Sample space:HHHTTT HHTTTH HTHTHT THHHTT Random variables:0 Head 1 Head 2 Heads 3 Heads 0 head:1 1 head:3 2 heads:3 3 heads:1 / 8 = / 8 = / 8 = 0.125

Binomial distribution

Poisson distribution

Normal distribution

 The center of the curve represents the mean, median, and mode.  The curve is symmetrical around the mean.  The tails meet the x-axis in infinity.  The curve is bell-shaped.  The total under the curve is equal to 1 (by definition). Normal distribution

Standard normal distribution 1.96

x 1 – x SD z-scores

Zwei Kandidaten haben an zwei unterschiedlichen Sprachtests teilgenommen. Kandidat A hat 121 Punkte erzielt, Kandidat B hat 177 Punkte erzielt. Im ersten Test (an dem Kandidat A teilgenommen hat) lag der Mittelwert bei 92 und die Standardabweichung bei 14; im zweiten Test (an dem Kandidat B teilgenommen hat) lag der Mittelwert bei 143 und die Standardabweichung bei 21. Welcher der beiden Kandidaten hat besser abgeschnitten (im Vergleich zu allen übrigen Kandidaten)? Z A = 121 – 92 / 14 = 2.07 Z B = 177 – 143 / 21 = 1.62

Central limit theorem

6, 2, 5, 6, 2, 3, 1, 6, 1, 1, 4, 6, 6, 2, 2, 1, 1, 5, 1, 3 = 2.64

X1X1 X2X2 X3X3 X4X4 M Sample Central limit theorem

X1X1 X2X2 X3X3 X4X4 M Sample Sample Central limit theorem

X1X1 X2X2 X3X3 X4X4 M Sample Sample Sample Central limit theorem

X1X1 X2X2 X3X3 X4X4 M Sample Sample Sample Sample Central limit theorem

X1X1 X2X2 X3X3 X4X4 M Sample Sample Sample Sample Sample Central limit theorem

= Mean of sample mean

The sample means are normally distributed (even if the phenomenon in the parent population is not normally distributed). Central limit theorem

 Der Mittelwert der individuellen Mittelwerte nähert sich dem Mittelwert in der wahren Population an.  Die Mittelwerte der Stichproben ist normalverteilt, selbst wenn das Phänomen, das wir untersuchen, in der wahren Population nicht normalverteilt ist.  Alle parametrischen Tests nutzen die Tatsache, dass die Mittelwerte der Stichproben (ab einer bestimmten Anzahl von Stichproben) normalverteilt sind. Central limit theorem

population

sample

population sample mean of this sample

population sample mean of this sample distribution of many sample means

How many samples do you need to assume that the mean of the sample means is normally distributed? Are your data normally distributed?

 The distribution in the parent population (normal, slightly skewed, heavily skewed).  The number of observations in the individual sample.  The total number of individual samples. Are your data normally distributed?

Confidence intervals

Confidence intervals indicate a range within which the mean (or other parameters) of the true population is located given the values of your sample and assuming a particular degree of certainty. Confidence intervals

 The mean of the sample means  The SDs of the sample means, i.e. the standard error  The degree of certainty with which you want to state the estimation Confidence intervals

 (x n – x) 2 N- 1 Standard deviation

SamplesMean  8.3 / 5 = 1.66 (mean) Standard error

SamplesMeanIndividual means – Mean of means – – – – – 1.66  8.3 / 5 = 1.66 (mean) Standard error

SamplesMeanIndividual means – Mean of means – – – – – –  8.3 / 5 = 1.66 (mean) Standard error

SamplesMeanIndividual means – Mean of means squared – – – – – –  8.3 / 5 = 1.66 (mean) Standard error

SamplesMeanIndividual means – Mean of means squared – – – – – –  8.3 / 5 = 1.66 (mean)  Standard error

= Standard error

[degree of certainty]  [standard error] = x [sample mean] +/–x = confidence interval Confidence intervals

95% degree of certainty = 1.96 [z-score] Confidence interval of the first sample (mean = 1.5): 1.96  = / = 0.97–2.03 We can be 95% certain that the population mean is located in the range between 0.97 and Confidence intervals

SD N Confidence intervals

What is the 95% confidence interval of the following sample: 2, 5, 6, 7, 10, 12? SD: (2-7) 2 + (5-7) 2 + (6-7) 2 + (7-7) 2 + (10-7) 2 + (12-7) Standard error:3.58 /  6 = 1.46 Mean: 7 = 3.58 Confidence I.:1.46  1.96 = /– 2.86 = 4.14 – 9.86

Confidence intervals