Binomne slučajne varijable
U ovom ćemo postu raspravljati o binomnim slučajnim varijablama.
Preduvjet: Slučajne varijable
Specifična vrsta diskretan slučajna varijabla koja broji koliko se često određeni događaj pojavljuje u fiksnom broju pokušaja ili pokušaja.
Da bi varijabla bila binomna slučajna varijabla SVI sljedeći uvjeti moraju biti ispunjeni:
- Postoji fiksni broj ispitivanja (fiksna veličina uzorka).
- U svakom pokusu se događaj od interesa ili dogodi ili ne.
- Vjerojatnost pojavljivanja (ili ne) je ista za svaki pokušaj.
- Suđenja su neovisna jedna o drugoj.
Matematičke oznake
n = number of trials
p = probability of success in each trial
k = number of success in n trials
Sada pokušavamo pronaći vjerojatnost k uspjeha u n pokušaja.
Ovdje je vjerojatnost uspjeha u svakom pokušaju p neovisna o drugim pokušajima.
Dakle, prvo biramo k pokusa u kojima će biti uspjeh, au preostalih n-k pokusa bit će neuspjeh. Broj načina za to je
Budući da je svih n događaja neovisno, stoga je vjerojatnost k uspjeha u n pokušaja ekvivalentna množenju vjerojatnosti za svaki pokušaj.
Ovdje je njegovih k uspjeha i n-k neuspjeha. Dakle, vjerojatnost za svaki način postizanja k uspjeha i n-k neuspjeha je
Stoga je konačna vjerojatnost
(number of ways to achieve k success
and n-k failures)
*
(probability for each way to achieve k
success and n-k failure)
Tada je vjerojatnost binomne slučajne varijable dana kao:
Neka je X binomna slučajna varijabla s brojem pokušaja n i vjerojatnošću uspjeha u svakom pokušaju p.
Očekivani broj uspjeha dat je pomoću
E[X] = np
Varijacija broja uspjeha dana je s
Var[X] = np(1-p)
Primjer 1 : Razmotrite nasumični eksperiment u kojem se novčić s pristranošću (vjerojatnost glave = 1/3) baca 10 puta. Odredite vjerojatnost da će broj glava koje se pojave biti 5.
rješenje:
Let X be binomial random variable
with n = 10 and p = 1/3
P(X=5) = ?
Ovdje je implementacija za isto
C++
Java// C++ program to compute Binomial Probability #include#include using namespace std ; // function to calculate nCr i.e. number of // ways to choose r out of n objects int nCr ( int n int r ) { // Since nCr is same as nC(n-r) // To decrease number of iterations if ( r > n / 2 ) r = n - r ; int answer = 1 ; for ( int i = 1 ; i <= r ; i ++ ) { answer *= ( n - r + i ); answer /= i ; } return answer ; } // function to calculate binomial r.v. probability float binomialProbability ( int n int k float p ) { return nCr ( n k ) * pow ( p k ) * pow ( 1 - p n - k ); } // Driver code int main () { int n = 10 ; int k = 5 ; float p = 1.0 / 3 ; float probability = binomialProbability ( n k p ); cout < < 'Probability of ' < < k ; cout < < ' heads when a coin is tossed ' < < n ; cout < < ' times where probability of each head is ' < < p < < endl ; cout < < ' is = ' < < probability < < endl ; } Python3// Java program to compute Binomial Probability import java.util.* ; class GFG { // function to calculate nCr i.e. number of // ways to choose r out of n objects static int nCr ( int n int r ) { // Since nCr is same as nC(n-r) // To decrease number of iterations if ( r > n / 2 ) r = n - r ; int answer = 1 ; for ( int i = 1 ; i <= r ; i ++ ) { answer *= ( n - r + i ); answer /= i ; } return answer ; } // function to calculate binomial r.v. probability static float binomialProbability ( int n int k float p ) { return nCr ( n k ) * ( float ) Math . pow ( p k ) * ( float ) Math . pow ( 1 - p n - k ); } // Driver code public static void main ( String [] args ) { int n = 10 ; int k = 5 ; float p = ( float ) 1.0 / 3 ; float probability = binomialProbability ( n k p ); System . out . print ( 'Probability of ' + k ); System . out . print ( ' heads when a coin is tossed ' + n ); System . out . println ( ' times where probability of each head is ' + p ); System . out . println ( ' is = ' + probability ); } } /* This code is contributed by Mr. Somesh Awasthi */C## Python3 program to compute Binomial # Probability # function to calculate nCr i.e. # number of ways to choose r out # of n objects def nCr ( n r ): # Since nCr is same as nC(n-r) # To decrease number of iterations if ( r > n / 2 ): r = n - r ; answer = 1 ; for i in range ( 1 r + 1 ): answer *= ( n - r + i ); answer /= i ; return answer ; # function to calculate binomial r.v. # probability def binomialProbability ( n k p ): return ( nCr ( n k ) * pow ( p k ) * pow ( 1 - p n - k )); # Driver code n = 10 ; k = 5 ; p = 1.0 / 3 ; probability = binomialProbability ( n k p ); print ( 'Probability of' k 'heads when a coin is tossed' end = ' ' ); print ( n 'times where probability of each head is' round ( p 6 )); print ( 'is = ' round ( probability 6 )); # This code is contributed by mitsJavaScript// C# program to compute Binomial // Probability. using System ; class GFG { // function to calculate nCr // i.e. number of ways to // choose r out of n objects static int nCr ( int n int r ) { // Since nCr is same as // nC(n-r) To decrease // number of iterations if ( r > n / 2 ) r = n - r ; int answer = 1 ; for ( int i = 1 ; i <= r ; i ++ ) { answer *= ( n - r + i ); answer /= i ; } return answer ; } // function to calculate binomial // r.v. probability static float binomialProbability ( int n int k float p ) { return nCr ( n k ) * ( float ) Math . Pow ( p k ) * ( float ) Math . Pow ( 1 - p n - k ); } // Driver code public static void Main () { int n = 10 ; int k = 5 ; float p = ( float ) 1.0 / 3 ; float probability = binomialProbability ( n k p ); Console . Write ( 'Probability of ' + k ); Console . Write ( ' heads when a coin ' + 'is tossed ' + n ); Console . Write ( ' times where ' + 'probability of each head is ' + p ); Console . Write ( ' is = ' + probability ); } } // This code is contributed by nitin mittal.PHP< script > // Javascript program to compute Binomial Probability // function to calculate nCr i.e. number of // ways to choose r out of n objects function nCr ( n r ) { // Since nCr is same as nC(n-r) // To decrease number of iterations if ( r > n / 2 ) r = n - r ; let answer = 1 ; for ( let i = 1 ; i <= r ; i ++ ) { answer *= ( n - r + i ); answer /= i ; } return answer ; } // function to calculate binomial r.v. probability function binomialProbability ( n k p ) { return nCr ( n k ) * Math . pow ( p k ) * Math . pow ( 1 - p n - k ); } // driver program let n = 10 ; let k = 5 ; let p = 1.0 / 3 ; let probability = binomialProbability ( n k p ); document . write ( 'Probability of ' + k ); document . write ( ' heads when a coin is tossed ' + n ); document . write ( ' times where probability of each head is ' + p ); document . write ( ' is = ' + probability ); // This code is contributed by code_hunt. < /script>Izlaz:
Probability of 5 heads when a coin is tossed 10 times where probability of each head is 0.333333
is = 0.136565Napravi kviz
