Compute Poisson Probabilities in C

Introduction

In this lab, you will learn how to compute Poisson probabilities in C. The Poisson distribution is a probability distribution that expresses the likelihood of a given number of events occurring in a fixed interval of time or space. You will first learn how to read the lambda (λ) parameter and k value, and then implement the Poisson probability formula to calculate the probability of exactly k events occurring in a given interval.

The lab covers the step-by-step process of writing a C program to compute Poisson probabilities, including handling user input, implementing the Poisson probability formula, and printing the calculated probability.

Skills Graph

Read λ (lambda) and k

In this step, you'll learn how to read the lambda (λ) parameter and k value for calculating Poisson probabilities in C. The Poisson distribution is a probability distribution that expresses the likelihood of a given number of events occurring in a fixed interval of time or space.

First, let's create a new C file to implement our Poisson probability calculation:

cd ~/project
nano poisson_prob.c

Now, add the following code to read lambda and k values:

#include <stdio.h>
#include <math.h>

int main() {
    double lambda;
    int k;

    printf("Enter lambda (average number of events): ");
    scanf("%lf", &lambda);

    printf("Enter k (number of occurrences): ");
    scanf("%d", &k);

    printf("Lambda (λ): %.2f\n", lambda);
    printf("k: %d\n", k);

    return 0;
}

Example output:

Enter lambda (average number of events): 3.5
Enter k (number of occurrences): 2
Lambda (λ): 3.50
k: 2

Let's break down the code:

• We use double for lambda to allow decimal values
• scanf() is used to read user input for lambda and k
• %lf format specifier is used for double (long float) values
• %d format specifier is used for integer values
• We print the entered values to confirm input

Compile the program:

gcc -o poisson_prob poisson_prob.c -lm

Run the program:

./poisson_prob

P(X=k)=e^(-λ)*λ^k/k!

In this step, you'll implement the Poisson probability formula to calculate the probability of exactly k events occurring in a given interval.

Open the previous file to modify the code:

cd ~/project
nano poisson_prob.c

Update the code to include the Poisson probability calculation:

#include <stdio.h>
#include <math.h>

// Function to calculate factorial
double factorial(int n) {
    if (n <= 1) return 1;
    double result = 1;
    for (int i = 2; i <= n; i++) {
        result *= i;
    }
    return result;
}

// Calculate Poisson probability
double poisson_probability(double lambda, int k) {
    double numerator = pow(lambda, k) * exp(-lambda);
    double denominator = factorial(k);
    return numerator / denominator;
}

int main() {
    double lambda;
    int k;

    printf("Enter lambda (average number of events): ");
    scanf("%lf", &lambda);

    printf("Enter k (number of occurrences): ");
    scanf("%d", &k);

    double probability = poisson_probability(lambda, k);

    printf("Lambda (λ): %.2f\n", lambda);
    printf("k: %d\n", k);
    printf("Probability P(X = %d): %.4f\n", k, probability);

    return 0;
}

Compile the program:

gcc -o poisson_prob poisson_prob.c -lm

Example output:

Enter lambda (average number of events): 3.5
Enter k (number of occurrences): 2
Lambda (λ): 3.50
k: 2
Probability P(X = 2): 0.1674

Key points in the code:

• factorial() function calculates k!
• poisson_probability() implements the Poisson probability formula
• pow() calculates λ^k
• exp() calculates e^(-λ)
• Result is probability of exactly k events occurring

Run the program:

./poisson_prob

Print the Probability

In this final step, you'll enhance the Poisson probability calculation program by adding more detailed probability printing and formatting.

Open the previous file to modify the code:

cd ~/project
nano poisson_prob.c

Update the code to improve probability presentation:

#include <stdio.h>
#include <math.h>

// Function to calculate factorial
double factorial(int n) {
    if (n <= 1) return 1;
    double result = 1;
    for (int i = 2; i <= n; i++) {
        result *= i;
    }
    return result;
}

// Calculate Poisson probability
double poisson_probability(double lambda, int k) {
    double numerator = pow(lambda, k) * exp(-lambda);
    double denominator = factorial(k);
    return numerator / denominator;
}

int main() {
    double lambda;
    int k;

    printf("Poisson Probability Calculator\n");
    printf("------------------------------\n");

    printf("Enter lambda (average number of events): ");
    scanf("%lf", &lambda);

    printf("Enter k (number of occurrences): ");
    scanf("%d", &k);

    double probability = poisson_probability(lambda, k);

    printf("\nCalculation Results:\n");
    printf("Lambda (λ): %.2f\n", lambda);
    printf("k: %d\n", k);
    printf("Probability P(X = %d): %.4f\n", k, probability);
    printf("Percentage: %.2f%%\n", probability * 100);

    return 0;
}

Compile the program:

gcc -o poisson_prob poisson_prob.c -lm

Example output:

Poisson Probability Calculator
------------------------------
Enter lambda (average number of events): 3.5
Enter k (number of occurrences): 2

Calculation Results:
Lambda (λ): 3.50
k: 2
Probability P(X = 2): 0.1674
Percentage: 16.74%

Key improvements:

• Added a title for the calculator
• Included percentage representation
• Improved output formatting
• More descriptive output sections

Run the program:

./poisson_prob

Summary

In this lab, you learned how to read the lambda (λ) parameter and k value for calculating Poisson probabilities in C. You then implemented the Poisson probability formula to calculate the probability of exactly k events occurring in a given interval. The Poisson distribution is a probability distribution that expresses the likelihood of a given number of events occurring in a fixed interval of time or space. You also learned how to create a factorial function to support the Poisson probability calculation.

The key steps covered in this lab include reading the input values, calculating the Poisson probability using the formula P(X=k)=e^(-λ)*λ^k/k!, and printing the resulting probability.