Lecture notes Statistics The Practice of Statistics for Business and Economics, ISBN: 9781319154127
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Course
Statistics
Institution
Rijksuniversiteit Groningen (RuG)
Book
The Practice of Statistics for Business and Economics
Lecture notes study book The Practice of Statistics for Business and Economics of David Moore, George McCabe - ISBN: 9781319154127, Edition: 4, Year of publication: - (recap + lectures)
A random phenomenon has outcomes that we cannot predict with certainty but that have a regular
distribution in very many repetitions. We cannot predict.
The probability of an event is the proportion of times the event occurs in may repeated trials of a
random phenomenon.
➔ Probability describes only what happens in the long run.
Independence: if the outcome of one trial don’t influence the outcome of any other.
➔ Replacement: independent.
𝑃(𝐵|𝐴) = 𝑃(𝐵)
Probability model; a description of a random phenomenon in the language of mathematics.
When we toss a coin. The description of coin tossing has 2 parts:
1. A list of possible outcomes. (Sample space)
2. A probability for each outcome. (Assignment of probabilities P)
Sample spaces (S); set of list of all possible outcomes of a random process.
𝑆 = {ℎ𝑒𝑎𝑑𝑠, 𝑡𝑎𝑖𝑙𝑠}
Event; an outcome or a set of outcomes of a random phenomenon. An event is a subset of the
sample space.
Probability rules;
1.
0 ≤ 𝑃(𝐴) ≤ 1
2. All possible outcomes of the sample space together must have probability 1.
3. The additional rule for disjoint events:
The additional rule holds if A and B are disjoint but not otherwise.
4. The complement rule: the probability that an event does not occur is 1 minus the probability
that the event does occur.
𝑃(𝑛𝑜𝑡 𝐴) = 𝑃(𝐴𝑐 ) = 1 − 𝑃(𝐴)
, 5. Multiplication rule for independent events: This multiplication rule holds if A and B are
independent but not otherwise.
➔ Only when independent.!
Venn diagram showing the complement Ac of an event A:
➔ The complement consists of all outcomes that are not in A:
Benford’s law;
What is the probability that the leftmost digit (“first digit’’) of a multidigit financial number is 9?
Many of us would assume the probability to be 1/9. Surprisingly, this is often not the case for
legitimately reported financial numbers. It is a striking fact that the first digits of numbers in
legitimate records often follow a distribution known as Benford’s law.
Equally likely outcomes: If a random phenomenon has k possible outcomes, all equally likely, then
each individual outcome has probability 1/k.
➔ If event A is independent, Ac is also independent, and Ac is independent of B.
0.2% (0.002)of the results are false positive.
If all of 150 are free of illegal drugs. The probability of a negative result is 1 – 0.02 = 0.998
The probability of at least one false-positive among the 150 people tested is:
, = 1 − 0.998150
= 1 − 0.741 = 0.259
4.3; General probability rules
The union of any collection of events is the event that at least one of the collections occurs.
i.e. P (A or B)
➔ For two events A and B, the union is
the event {A or B} that A or B or both
occur.
when the events are joint:
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) 𝑃(𝐵)
Above only when independent
Conditional probability P(B|A): gives the probability of one event under the condition that we know
another event has occurred. (requirement: if denominator > 0)
➔ Probability that B occurs, given the information that A occurs.
➔ You can read the bar u as “given the information that.
➔ Not-Independent events.
The intersection of any collection of events is the event that all the events occur.
➔ Contains all outcomes that are in both A and B, but not outcomes in A alone or B alone.
Summary general rules chapter 4.2:
Additional rule:
➔ When they are joint.
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
Multiplication rule:
➔ When they are not independent.
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) 𝑃(𝐵|𝐴)
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