Which of the following random variables is geometric?
In the field of probability and statistics, random variables play a crucial role in modeling various phenomena. One such type of random variable is the geometric distribution, which is widely used to describe the number of trials needed to achieve the first success in a sequence of independent and identically distributed Bernoulli trials. This article aims to explore which of the given random variables is geometric and discuss its properties and applications.
The geometric distribution is characterized by its probability mass function (PMF), which is defined as follows:
\[ P(X = k) = (1-p)^{k-1} \cdot p \]
where \( X \) represents the random variable, \( k \) is the number of trials until the first success, and \( p \) is the probability of success in each trial.
Now, let’s analyze the given random variables to determine which one follows a geometric distribution:
1. The number of times a coin is tossed until a head appears.
2. The number of customers arriving at a store in a given hour.
3. The number of rolls of a fair six-sided die until a six is rolled.
4. The number of emails received in a day.
To identify the geometric distribution among these random variables, we need to consider the conditions that must be met:
– The random variable represents the number of trials until the first success.
– Each trial is independent and identically distributed.
– The probability of success remains constant throughout the trials.
Let’s evaluate each option based on these criteria:
1. The number of times a coin is tossed until a head appears: This random variable is geometric because it represents the number of trials until the first success (a head) in a sequence of independent and identically distributed coin tosses. The probability of success (getting a head) remains constant at \( p = 0.5 \).
2. The number of customers arriving at a store in a given hour: This random variable is not geometric because it represents the total number of customers arriving in an hour, not the number of trials until the first customer arrives. The number of trials is not defined, and the independence and identical distribution of trials cannot be guaranteed.
3. The number of rolls of a fair six-sided die until a six is rolled: This random variable is geometric because it represents the number of trials until the first success (rolling a six) in a sequence of independent and identically distributed die rolls. The probability of success (rolling a six) remains constant at \( p = \frac{1}{6} \).
4. The number of emails received in a day: This random variable is not geometric because it represents the total number of emails received in a day, not the number of trials until the first email arrives. The number of trials is not defined, and the independence and identical distribution of trials cannot be guaranteed.
In conclusion, the random variables that follow a geometric distribution are:
1. The number of times a coin is tossed until a head appears.
3. The number of rolls of a fair six-sided die until a six is rolled.
The geometric distribution has numerous applications in various fields, such as queuing theory, reliability engineering, and biological sciences. Understanding the properties and characteristics of this distribution can help us model and analyze real-world scenarios more effectively.
