Probabilitas 08 | Belajar Distribusi Geometrik & Distribusi Poisson | Belajar Probabilitas Dasar

Introduction

In this tutorial, we will explore two important concepts in basic probability: Geometric Distribution and Poisson Distribution. Understanding these distributions is vital for analyzing random events and can be applied in various fields such as statistics, engineering, and data science. This guide provides a step-by-step breakdown of each distribution to help you grasp their applications and calculations.

Step 1: Understanding Geometric Distribution

Geometric distribution models the number of trials required for the first success in a series of independent Bernoulli trials (trials with two possible outcomes: success or failure).

Key Characteristics

• Definition: The probability of having the first success on the k-th trial.

• Formula: The probability mass function is given by:

  [ P(X = k) = (1 - p)^{k-1} \cdot p ]

  where:

  • ( p ) is the probability of success on each trial.
  • ( k ) is the trial number on which the first success occurs.

Practical Application

• Example: If you flip a biased coin where the probability of getting heads (success) is 0.3, you can calculate the probability of getting the first head on the 4th flip.

Common Pitfalls

• Ensure that the trials are independent and that the probability of success remains constant for each trial.

Step 2: Understanding Poisson Distribution

Poisson distribution is used to model the number of events occurring within a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event.

Key Characteristics

• Definition: Models the number of events in a fixed interval.

• Formula: The probability mass function is given by:

  [ P(X = k) = \frac{\lambda^k \cdot e^{-\lambda}}{k!} ]

  where:

  • ( \lambda ) is the average number of events in the interval.
  • ( k ) is the actual number of events that occur.
  • ( e ) is Euler's number (approximately equal to 2.71828).

Practical Application

• Example: If a call center receives an average of 5 calls per hour, you can calculate the probability of receiving exactly 3 calls in the next hour.

Common Pitfalls

• Ensure that the events are independent and that the average rate of occurrence is known.

Conclusion

In summary, both Geometric and Poisson distributions are essential for understanding random processes in probability. The Geometric Distribution is focused on the number of trials until the first success, while the Poisson Distribution deals with the number of events in a given interval. To deepen your understanding, practice calculating probabilities using different values for ( p ) and ( \lambda ).

Next Steps

• Explore more examples and practice problems related to these distributions to reinforce your understanding.
• Review related topics such as binomial distribution for a broader perspective on probability.