Binomna razdioba

undergraduate thesis

• Title (english): Binomial distribution
• Author: Luka Gregurek
• Mentor: Erna Begović Kovač (mentor)
• Committee member: Erna Begović Kovač (predsjednik povjerenstva)
• Committee member: Miroslav Jerković (član povjerenstva)
• Committee member: Dajana Kučić Grgić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Chemical Engineering and Technology; Zagreb
• Defense date and country: 2021-09-22, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Svakom ishodu slučajnog pokusa možemo pridružiti specifičnu numeričku vrijednost, što bilježimo sa funkcijom koja svakom ishodu pridruži određeni realni broj. Tu funkciju nazivamo slučajna varijabla. Diskretna slučajna varijabla je slučajna varijabla čije vrijednosti čine konačan ili prebrojiv skup. Slučajna varijabla u kojoj imamo samo dva moguća ishoda nazivamo Bernoullijvom slučajnom varijablom. Binomna slučajna varijabla dobije se kao rezultat n manjih pokusa Bernoullijvog tipa. Ona ovisi o dva parametra, n i p, gdje je n broj Bernoullijevih pokusa, a p vjerojatnost uspjeha u svakom Bernoullijevom pokusu. Binomna slučajna varijabla označava se 𝑋~𝐵(𝑛,𝑝). Funkcija gustoće vjerojatnosti slučajna varijable p(x) predstavlja vjerojatnost da je vrijednost slučajne varijable X jednaka broju x, tj. 𝑝(𝑥)=𝑃(𝑋=𝑥). Za slučajnu varijablu s binomnom distribucijom vrijedi 𝑃(𝑋=𝑥)=(𝑛𝑥)𝑝𝑥∗(1−𝑝)𝑛−𝑥. Ako takve vjerojatnosti želimo promatrati kumulativno, dolazimo do funkcije distribucije vjerojatnosti F(X) koja se definira kao 𝐹(𝑥)=𝑃(𝑋≤𝑥). Ukoliko želimo znati gdje je funkcija gustoće centrirana koristimo formulu za očekivanje E(X). Ako slučajna varijabla X ima binomnu distribuciju, vrijedi 𝐸(𝑋)=𝑛𝑝. Veličine koje daju procjenu odstupanja od očekivanja su varijanca i standardna devijacija. Za binomnu slučajnu varijablu formula za varijancu je 𝑉(𝑋)=𝑛𝑝(1−𝑝), a za standardnu devijaciju σ𝑋=√𝑉𝐴𝑅(𝑋).

Abstract (english)

To each outcome of a random experiment we can assign a specific numerical value. This is recorded by a function that assigns a certain real number to each outcome. This function is called a random variable. A discrete random variable is a random variable with values in a finite or a countable set. A random variable witch only two possible outcomes is called a Bernoulli random variable. A binomial random variable is obtained as a result of n smaller experiments of the Bernoulli type. It depends on two parameters, n and p, where n is the number of Bernoulli experiments, and p is the probability of success in each Bernoulli experiment. The binomial random variable is denoted by X~B(n, p). The density function of a random variable p(x) represents the probability that the value of the random variable X is equal to the number x, 𝑝(𝑥)=𝑃(𝑋=𝑥). For a random variable with binomial distribution, it holds 𝑃(𝑋=𝑥)=(𝑛𝑥)𝑝𝑥∗(1−𝑝)𝑛−𝑥. If we want to observe such probabilities cumulatively, we use the probability distribution function F(x). It is defined as 𝐹(𝑥)=𝑃(𝑋≤𝑥). If we want to know where the density function is centered, we use the expectation formula E(X). If the random variable X has binomial distribution, it holds 𝐸(𝑋)=𝑛𝑝. The values that give an estimate of the deviation from the expected value are variance and standard deviation. For a binomial random variable, the formula for the variance is 𝑉(𝑋)=𝑛𝑝(1−𝑝), and for the standard deviation σ𝑋=√𝑉𝐴𝑅(𝑋).

• Keywords: slučajne varijable; binomna distribucija; funkcija gustoće vjerojatnosti; funkcija distribucije vjerojatnosti; očekivanje; varijanca; standardna devijacija.
• Keywords (english): random variable; binomial distribution; probability density function; probability distribution; expected value; variance; standard deviation
• Language: croatian
• URN:NBN: urn:nbn:hr:149:012930
• Study programme: Title: Chemical Engineering - Undergraduate study; Study programme type: university; Study level: undergraduate; Academic / professional title: sveučilišni/a prvostupnik/ prvostupnica (baccalaureus/baccalaurea) inženjer/inženjerka kemijskog inženjerstva (sveučilišni/a prvostupnik/ prvostupnica (baccalaureus/baccalaurea) inženjer/inženjerka kemijskog inženjerstva)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2023-03-29 11:43:53