Probability space, probability function, sample space, event
Conditional probability
In the French Open final, Nadal will play the winner of the semi-final between Federer and Davydenko. The chance that Nadal can beat Federer is estimated at 51%, while the chance that Nadal can beat Davydenko is estimated at 80%.
Independent events
The Inclusion-Exclusion Formula
We want to calculate the probability P(B) of the event that all four suits appear among the 5 selected cards. To this end, let A1 be the event that none of the chosen cards are spades.
Binomial coefficients
For example, the number of subsets with 5 elements (poker hands) of a set with 52 elements (a deck of cards) is equal to 52. An easy way to remember the binomial coefficients is to arrange them in Pascal's triangle where each number is equal to the sum of the numbers immediately above:.
Multinomial coefficients
P(X =x): the probability that X takes on the value x∈R, P(X < x): the probability that accepts, etc.
The distribution function
Discrete random variables, point probabilities
Continuous random variables, density function
Continuous random variables, distribution function
Independent random variables
Roll a red die and a black die and consider the random variables X: the number of cups on the red die,. In contrast, X and Dice are not independent since information about Z provides information about X (if, for example, Z has the value 10, then X must have one of the values 4, 5, and 6).
Random vector, simultaneous density, and distribution function
Variance and standard deviation of random variables
Example (computation of expected value, variance, and standard deviation)
Estimation of expected value µ and standard deviation σ by eye
Addition and multiplication formulae for expected value and variance
Covariance and correlation coefficient
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The Law of Large Numbers
The Central Limit Theorem
Example (distribution functions converge to Φ)
Mean value
Empirical variance and empirical standard deviation
The larger the empirical standard deviations, the more "spread" the observations are around the mean valuex.¯.
Empirical covariance and empirical correlation coefficient
Significance probability and significance level
If P is greater than α, then H0 is accepted (and we say "H0 is accepted at significance level α" or "H0 cannot be rejected at significance level α").
Errors of type I and II
Example
Description
Point probabilities
Expected value and variance
Significance probabilities for tests in the binomial distribution
The normal approximation to the binomial distribution
Can the company now reject the manufacturer's claim that the probability of failure is at most one sixth.
Estimators
Assuming it is within two standard deviations of , we can conclude that it is between 10% and 50%.
Confidence intervals
So we can say with 95% certainty that between 52% and 72% of the electorate will vote for the Green Party in the next election.
Description
Point probabilities
Expected value and variance
Addition formula
Significance probabilities for tests in the Poisson distribution
Example (significant increase in sale of Skodas)
The binomial approximation to the Poisson distribution
The normal approximation to the Poisson distribution
Example (significant decrease in number of complaints)
Estimators
Confidence intervals
Description
Point probabilities and tail probabilities
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Expected value and variance
Description
Point probabilities and tail probabilities
Expected value and variance
The binomial approximation to the hypergeometrical distribution
The normal approximation to the hypergeometrical distribution
Parameters
Description
Point probabilities
Estimators
Description
Point probabilities
Expected value and variance
Estimators
Description
Density and distribution function
Expected value and variance
Description
Density and distribution function
The standard normal distribution
Properties of Φ
Estimation of the expected value µ
Estimation of the variance σ 2
Confidence intervals for the expected value µ
Confidence intervals for the variance σ 2 and the standard deviation σ
Addition formula
Student’s t distribution
In practice, the density is not used, but the distribution function is found in Table B.3. The graph below shows the density of this distribution with df = 1,2,3 degrees of freedom and the additional density ϕ(x) of the standard normal distribution.
Fisher’s F distribution
In practice, one does not use the density, but instead looks up the distribution function in table B.4. The significance probability now appears from the following table, where Φ is the distribution function of the standard normal distribution (table B.2).
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Example
The probability of significance is read from the table below where Φ is the distribution function of the standard normal distribution (Table B.3).
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Example (comparison of two expected values)
Two examples (comparison of mean values from three samples)
The assumption of normal distribution
Standardized residuals
Example (women with five children)
Test the hypothesis H0 that, at each birth, the probability of a boy is the same as the probability of a girl. We note that it is the figures of women with five children of the same sex that are extreme and make the statistic large.
Example (election)
By looking up in Table B.3 under the χ2 distribution withdf = 8−1 = 7 degrees of freedom, it is only seen that the significance probability is below 50%. So we have no statistical evidence to conclude that the popularity of the parties has changed.
Example (deaths in the Prussian cavalry)
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Standardized residuals
Example (students’ political orientation)
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Fisher’s exact test for 2 × 2 tables
The conditional probability of obtaining exactly the 2×2 table above, given that the row sums are R1 and R2, and that the column sums are S1 and S2, is It is then necessary that the 2×2 table be written in such a way that the observed numbers in the diagonal are greater than the expected numbers (this can always be achieved by switching the rows if necessary).
Example (Fisher’s exact test)
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Example
If we do not test H0 against a certain alternative hypothesis, the null hypothesis is rejected if the minimum := min{t+, t−} is extremely small. Conclusion: The test shows that the null hypothesis H0 must be rejected against the alternative hypothesis H1 at a significance level of 5%.
The normal approximation to Wilcoxon’s test for one set of observations
Wilcoxon’s test for two sets of observations
If we do not test H0 against some alternative hypothesis, the null hypothesis is rejected if it is the minimum. If it is less than or equal to the table value, H0 is rejected at significance levelα= 10%.
The normal approximation to Wilcoxon’s test for two sets of observations
Estimation of the parameters β 0 and β 1
The distribution of the estimators
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Predicted values y ˆ i and residuals e ˆ i
Estimation of the variance σ 2
Confidence intervals for the parameters β 0 and β 1
The determination coefficient R 2
Predictions and prediction intervals
Overview of formulae
Example
How to read the tables
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The standard normal distribution
Student’s t distribution (values x with F Student (x) = 0.600 etc.)
Fisher’s F distribution (values x with F Fisher (x) = 0.90)
Fisher’s F distribution (values x with F Fisher (x) = 0.95)
Fisher’s F distribution (values x with F Fisher (x) = 0.99)
Wilcoxon’s test for one set of observations
Wilcoxon’s test for two sets of observations, α = 5%
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