Case 1

Definition

Case 1 refers to a specific scenario in the context of the Geometric Distribution, which is a discrete probability distribution that models the number of trials required to obtain the first success in a series of independent Bernoulli trials with a constant probability of success.

5 Must Know Facts For Your Next Test

1. In Case 1, the Geometric Distribution is used to model the number of trials required to obtain the first success in a series of independent Bernoulli trials with a constant probability of success.
2. The probability of success in each Bernoulli trial is denoted by the parameter $p$, where $0 < p < 1$.
3. The probability mass function (PMF) of the Geometric random variable $X$ in Case 1 is given by $P(X = x) = p(1-p)^{x-1}$, where $x$ is a positive integer.
4. The expected value (mean) of the Geometric random variable in Case 1 is $E(X) = \frac{1}{p}$, and the variance is $Var(X) = \frac{1-p}{p^2}$.
5. Case 1 of the Geometric Distribution is often used to model various real-world phenomena, such as the number of attempts required to obtain the first success in a series of independent trials, the time until the first success in a continuous process, or the number of customers arriving at a service facility before the first customer is served.

Review Questions

• Explain the relationship between the Geometric Distribution and Bernoulli trials in the context of Case 1.
  • In Case 1 of the Geometric Distribution, the random variable $X$ represents the number of trials required to obtain the first success in a series of independent Bernoulli trials with a constant probability of success $p$. Each Bernoulli trial has only two possible outcomes: success or failure, and the probability of success remains the same across all trials. The Geometric Distribution models the number of trials needed to obtain the first success, which is a discrete random variable with a probability mass function that depends on the parameter $p$.
• Describe the properties of the probability mass function (PMF) for the Geometric random variable in Case 1.
  • The probability mass function (PMF) of the Geometric random variable $X$ in Case 1 is given by $P(X = x) = p(1-p)^{x-1}$, where $x$ is a positive integer and $0 < p < 1$. This PMF has the following properties: 1) The PMF is a decreasing function of $x$, meaning the probability of obtaining the first success on the $x$-th trial decreases as $x$ increases. 2) The sum of the probabilities for all possible values of $x$ is equal to 1, as the Geometric Distribution models the probability of obtaining the first success at some point. 3) The PMF is geometric in shape, with the probability of success on each trial being constant.
• Analyze the relationship between the expected value (mean) and variance of the Geometric random variable in Case 1, and explain the practical implications of these measures.
  • In Case 1 of the Geometric Distribution, the expected value (mean) of the random variable $X$ is $E(X) = \frac{1}{p}$, and the variance is $Var(X) = \frac{1-p}{p^2}$. The expected value represents the average number of trials required to obtain the first success, while the variance measures the spread or dispersion of the distribution. The practical implications of these measures are: 1) The expected value indicates the typical number of trials needed to obtain the first success, which is useful for planning and decision-making. 2) The variance provides information about the variability in the number of trials, which is important for understanding the uncertainty and risk associated with the process. Together, the mean and variance of the Geometric random variable in Case 1 give a comprehensive understanding of the distribution and its practical applications.

Related terms

Bernoulli Trial: A Bernoulli trial is a random experiment with only two possible outcomes, success or failure, and a constant probability of success.

Geometric Random Variable: A geometric random variable represents the number of trials required to obtain the first success in a series of independent Bernoulli trials with a constant probability of success.

Probability Mass Function (PMF): The probability mass function (PMF) is a function that gives the probability of each possible value of a discrete random variable.