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tanszek:oktatas:techcomm:statistical_properties

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tanszek:oktatas:techcomm:statistical_properties [2025/09/29 17:59] – [Frequency of Events] kneheztanszek:oktatas:techcomm:statistical_properties [2025/09/30 05:33] (current) – [Frequency of Events] knehez
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-====== Statistical properties ======+====== Statistical properties of Information ======
  
 ==== Event Space ==== ==== Event Space ====
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 The outcome of //experiments//, the //results of observations//, and the //status of systems// form a so-called '**event space**', where finite or infinite cardinality elementary events may occur. In probability theory, an __event space__ encompasses (includes) all possible outcomes of an experiment, such as flipping a coin, rolling a dice, or measuring the temperature of a system. The outcome of //experiments//, the //results of observations//, and the //status of systems// form a so-called '**event space**', where finite or infinite cardinality elementary events may occur. In probability theory, an __event space__ encompasses (includes) all possible outcomes of an experiment, such as flipping a coin, rolling a dice, or measuring the temperature of a system.
  
-These events may form sets. Because they can be sets, we may perform standard //set operations// (union, intersection, complements, etc.) on them. For example, the union of two events in an experiment may represent the occurrence of either event, while the intersection represents the simultaneous occurrence of both. When these operations occur, the event will carry information.+These events may form sets. Because they can be sets, we may perform standard //set operations// (union, intersection, complements, etc.) on them.  
 + 
 +For example: the __union of two events__ in an experiment may represent the //occurrence of either event//, while the __intersection__ represents the //simultaneous occurrence of both//. When these operations occur, the event will carry information.
  
 The __value of information__ related to these events can vary significantly based on everyday experience. For instance, knowing that a rare event has occurred (such as winning the lottery) typically provides more valuable information than learning about an event that occurs frequently or predictably. The __value of information__ related to these events can vary significantly based on everyday experience. For instance, knowing that a rare event has occurred (such as winning the lottery) typically provides more valuable information than learning about an event that occurs frequently or predictably.
  
-For example, if someone tells me that five of his numbers were drawn in the lottery, that information would be much more valuable than if they said only one number was drawn.+For example: if someone tells me that five of his numbers were drawn in the lottery, that information would be much more valuable than if they said only one number was drawn.
  
 As we observe the outcomes of these events over time, we may conclude that certain events exhibit stability in their __frequency of occurrence__. For example, when repeatedly flipping a coin, we expect the event of landing heads to occur approximately 50% of the time, given enough trials. This regularity in frequency forms the basis for probability theory, where events with stable frequencies are described as having predictable probabilities. As we observe the outcomes of these events over time, we may conclude that certain events exhibit stability in their __frequency of occurrence__. For example, when repeatedly flipping a coin, we expect the event of landing heads to occur approximately 50% of the time, given enough trials. This regularity in frequency forms the basis for probability theory, where events with stable frequencies are described as having predictable probabilities.
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 In the \( E_i \) event space, an event happened \(k_i\) times then the frequency of that given event may be calculated with the following formula:  In the \( E_i \) event space, an event happened \(k_i\) times then the frequency of that given event may be calculated with the following formula: 
  
-$$ freq_i=\frac{k_i}{k} $$+$$ freq_i=\frac{k_i}{k} = \frac{\text{number of favorable outcomes}}{\text{number of all possible outcomes}}$$
  
-This means that we divide the number of all events by the number (frequency) of that given event. In case of a large number of experiments this number will show us the probability of that event.+This means that we divide the number of that //given event// by the number of all //experiments//. In the case of a large number of experiments, this number will show us the probability of that event.
  
 $$ \lim_{k \to \infty} freq_i = \frac{k_i}{k} =  P(E_i) $$ $$ \lim_{k \to \infty} freq_i = \frac{k_i}{k} =  P(E_i) $$
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 {{:tanszek:oktatas:techcomm:pasted:20240826-164556.png}} {{:tanszek:oktatas:techcomm:pasted:20240826-164556.png}}
  
-The probability of every event will be 1/6 which can be calculated with the following formula. The rolls (of the dice) will form a so-called //full event system//. In the case of a //full event system// the sum of the probabilities (of each event) is 1 (by definition).+The probability of every event will be \(\frac{1}{6}\), which can be calculated with the following formula. The rolls (of the dice) will form a so-called //full event system//. In the case of a //full event system// the sum of the probabilities (of each event) is 1 (by definition).
  
 If one of the events happens then the others can not happen: If one of the events happens then the others can not happen:
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 Create sets containing all the possibilities.    Create sets containing all the possibilities.   
  
-Try the following c code here: https://www.onlinegdb.com/online_c_compiler and examine the results.+Try the following c code here: [[https://www.onlinegdb.com/online_c_compiler| online compiler]] and examine the results.
  
 <sxh c> <sxh c>
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 #include <time.h> #include <time.h>
  
-#define SIMULATIONS 1000+#define SIMULATIONS 10000
  
 // Function to simulate the birth of twins // Function to simulate the birth of twins
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     // Calculate and print the result     // Calculate and print the result
-    double probability = (double)at_least_one_girl / SIMULATIONS * 100; +    float probability = (float)at_least_one_girl / SIMULATIONS; 
-    printf("In %.2f%% of cases, there is at least one girl in the twin pair.\n", probability);+    printf("In %.2f%% of cases, there is at least one girl in the twin pair.\n", probability * 100.0f);
  
     return 0;     return 0;
tanszek/oktatas/techcomm/statistical_properties.1759168747.txt.gz · Last modified: 2025/09/29 17:59 by knehez