The mean of the geometric distribution is given by _____________.Question 3Answera.npb.1/pc.p

The mean of a geometric distribution is an important concept in probability and statistics. Let's break down the question and provide the answer clearly.

1. Break Down the Problem

   • We want to determine the mean of the geometric distribution. The parameters involved are $p$ (the probability of success) and $q$ (the probability of failure), where $q = 1 - p$.

2. Relevant Concepts

   • In probability theory, a geometric distribution models the number of trials needed to achieve the first success in repeated, independent Bernoulli trials. The mean (expected value) of a geometric distribution is given by the formula: $E(X) = \frac{1}{p}$

3. Analysis and Detail

   • Here, $E(X)$ represents the expected number of trials until the first success occurs, where $p$ is the probability of success. Therefore, if the probability of success is 0.5, the mean number of trials until the first success would be: $E(X) = \frac{1}{0.5} = 2$
   • This indicates that on average, you would expect to conduct 2 trials to achieve the first successful outcome.

4. Verify and Summarize

   • The formula $E(X) = \frac{1}{p}$ aligns with our understanding of the geometric probability distribution. It encapsulates the relationship between the probability of success and the expected number of trials. Thus, the correct answer for the mean of the geometric distribution is succinctly expressed using this formula.

Final Answer

The mean of the geometric distribution is given by: $E(X) = \frac{1}{p}$

Here, $p$ is the probability of success in each trial.