Waiting time refers to the length of time a person or object must wait before a certain event occurs. In the context of the Geometric Distribution, waiting time describes the number of trials or attempts required before the first success is achieved.
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The waiting time for the first success in a series of Bernoulli trials follows a Geometric distribution.
The Geometric distribution models the number of trials or attempts required before the first success occurs.
The probability of success on each trial is constant and denoted as 'p'.
The mean of the Geometric distribution is $\frac{1}{p}$, where 'p' is the probability of success on each trial.
The variance of the Geometric distribution is $\frac{1-p}{p^2}$.
Review Questions
Explain how the waiting time is related to the Geometric distribution.
In the context of the Geometric distribution, the waiting time refers to the number of trials or attempts required before the first success is achieved. The Geometric distribution models the number of Bernoulli trials needed to obtain the first success, where each trial has a constant probability of success 'p'. The waiting time, represented by the Geometric random variable, follows this distribution and describes the length of time a person or object must wait before the first successful outcome occurs.
Describe the relationship between the probability of success 'p' and the mean and variance of the Geometric distribution.
The probability of success 'p' on each Bernoulli trial is a key parameter of the Geometric distribution. The mean of the Geometric distribution is $\frac{1}{p}$, meaning the average waiting time before the first success is achieved is inversely proportional to the probability of success 'p'. Similarly, the variance of the Geometric distribution is $\frac{1-p}{p^2}$, which indicates that as the probability of success 'p' increases, the variance, and thus the spread of the distribution, decreases.
Analyze how the waiting time can be used to make inferences about the underlying Bernoulli trials in the Geometric distribution.
The waiting time, as modeled by the Geometric distribution, can provide valuable insights about the Bernoulli trials that generate the data. By observing the distribution of waiting times, one can make inferences about the probability of success 'p' on each trial. For example, if the waiting times tend to be shorter, it suggests a higher probability of success 'p' in the Bernoulli trials. Conversely, longer waiting times imply a lower probability of success 'p'. This relationship between the waiting time and the underlying probability of success can be used to draw conclusions about the characteristics of the Bernoulli trials in the Geometric distribution.
Related terms
Geometric Random Variable: A discrete random variable that represents the number of trials or attempts required to obtain the first success in a series of independent Bernoulli trials, where each trial has only two possible outcomes: success or failure.
A single experiment or observation that has only two possible outcomes, typically labeled as 'success' and 'failure,' with a constant probability of success on each trial.