2010 - Essentials of Statistics

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Good lecture!

Some random paragraphs :)
The point probabilities ...
In contrast, X and Z are not independent since information about Z yields information  about X (if, for example, Z has the value 10, then X necessarily has one of the values 4, 5 and 6)....
3.7 Random vector, simultaneous density, and distribution fonction ...
If X1, ... , Xn are random variables defined on the same probability space (n, P) we call X = (X1 , ... , Xn ) an (n-dimensional) random vector. It is a map
X: [1 _, IRn .
The simultaneous (n-dimensional) distribution fonction is the fonction F : IRn _, [0, 1] given by .......
4.4 Estimation of expected value µ and standard deviation a by eye
If the density fonction (or a pin diagram showing the point probabilities) of a random variable is given, one can estimate µ and cr by eye. The expected value µ is approximately the "centre of mass" of the distribution, and the standard deviation cr has a size such that more or less two thirds of the "probability mass" lie in the interval µ±cr....
Remember: A positive correlation coefficient implies that normally Xis large when Y large, and vice versa. A negative correlation coefficient implies that normally X is small when Y is large, and vice versa.. ...
 It is assumed that a2 is positive. Denote by S􀀎 the normed sums ..........
Understanding of the empirical correlation coefficient. ..then r will be equal or close to -1....
7. Statistical hypothesis testing
7.1 Null hypothesis and alternative hypothesis
A statistical test is a procedure that leads to either acceptance or rejection of a null hypothesis Ho given in advance. Sometimes Ho is tested against an explicit alternative hypothesis H1.
At the base of the test lie one or more observations. ....
7.2 Significance probability and significance level
One computes the significance probability P, that is the probability- if Ho is true - of obtaining an observation .
Often, one chooses a significance level a: in advance, typically a: = 5%. One then rejects Ho if Pis smaller than a: (and one says, "Ho is rejected at significance level a:").... Thus, the strength should be as great as possible.

7.4 Example
Suppose we wish to investigate whether a certain <lice is fair. By "fair" we here only understand that the probability p of a six is 1/6. . Can the company reject the manufacturer's claim on this background?
SOLUTION. We test the null hypothesis Ho : p = 1/6 against the alternative hypothesis H1 p > 1/6. The significance probability can be computed as P(X 2". 6) = 10.2% (see e.g. t
8.6 The normal approximation to the binomial distribution
If the parameter n (the number of tries) is large, a binomially distributed random variable X will be approximately normally distributed with expected value µ = np and standard deviation CT = y'np(i. Therefore, the point probabilities are approximately
EXAMPLE (continuation of the example in section 8.5). After 2 weeks the machine has produced 200 chips of which 46 are defective. Can the company now reject the manufacturer's daim that the probability of defects is at most one six th? "

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2010 - Essentials of Statistics 

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