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Cumulative Probabilities in Geometric Distributions: A Summary

Geometric distributions model situations where you face repeated trials with a constant probability of success on each attempt. The key question often involves cumulative probabilities: what’s the likelihood of achieving success within a certain number of trials? Here’s how to tackle it:

1. Understand the Players:

  • p: Probability of success in each trial (must be between 0 and 1).
  • q: Probability of failure (1 – p).
  • X: Random variable representing the number of trials needed for the first success.

2. The Formula:

The cumulative distribution function (CDF) tells you the probability of obtaining success in X ≤ x trials:

P(X ≤ x) = 1 - (1 - p)^x
  • This formula considers all outcomes up to and including the xth trial.

3. Interpreting the Result:

  • The value of P(X ≤ x) lies between 0 and 1.
  • As x increases, P(X ≤ x) approaches 1 (certainty of success eventually).
  • For a specific x, the result represents the chance of success within those x trials.

4. Special Cases:

  • P(X = 1): The probability of success on the very first trial, which equals p.
  • P(X > x): The opposite of P(X ≤ x), calculated as 1 – P(X ≤ x).

5. Beyond Formulas:

  • Calculators and statistical software can efficiently compute cumulative probabilities.
  • Visualizing the geometric distribution through graphs can aid understanding.

Remember:

  • The formula assumes independent trials with constant success probability.
  • Adapt the approach if dealing with more complex scenarios involving additional parameters or dependencies.

In a geometric distribution, P(X > x) represents the probability that it takes more than x trials to achieve the first success. It’s essentially the opposite of the cumulative probability P(X ≤ x), which we discussed earlier.

Here’s how you can find P(X > x):

1. Leverage the Relationship:

P(X > x) is directly related to P(X ≤ x) through the following equation:

P(X > x) = 1 - P(X ≤ x)

2. Apply the CDF Formula:

Substitute the formula for P(X ≤ x) from the previous explanation:

P(X > x) = 1 - [1 - (1 - p)^x]

3. Simplify:

Combine the terms:

P(X > x) = (1 - p)^x

This formula directly gives you the probability of needing more than x trials for success in the geometric distribution.

Example:

Suppose the probability of success in each trial is p = 0.4 and you’re interested in P(X > 3).

P(X > 3) = (1 - 0.4)^3 = 0.216

Therefore, there’s a 38.4% chance that it will take more than 3 trials for the first success to occur in this scenario.

Mark

Breton Expat in Suzhou #guingamp #suzhou

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