Geometric Distribution Examples
Some of the geometric distribution real-life examples are given below:
A person is looking for a job that is both challenging and satisfying. What is the probability that he will drop out zero times, one time, two times, and so on unless he gets his new job?
Examine a couple who are planning to conceive a child and they will proceed with babies unless it is a girl. What is the probability that the couple has zero boys, one boy, two boys, and so on unless a girl is delivered?
A pharmaceutical company is planning to design a new drug to treat a certain disease that will have minimum side effects. What is the probability that zero drugs failed the test, one drug failed the test, two drugs failed the test, and so on unless they come up with the newly designed ideal drug.?
Mean of Geometric Distribution
Each probability distribution has its particular formula for mean and variance of the random variable x. The mean of the expected value of x determines the weighted average of all possible values for x. For a mean of geometric distribution E(X) or μ is derived by the following formula.
E(Y) = μ = 1/P
Solved Examples
1. Find the probability density of geometric distribution if the value of p is 0.42; x = 1,2,3 and also calculate the mean and variance.
Solution:
Given that p = 0.42 and the value of x = 1, 2, 3
The formula of probability density of geometric distribution is
P(x) = p (1-p) x-1; x =1, 2, 3
P(x) = 0; otherwise
P(x) = 0.42 (1- 0.42)
P(x) = 0; Otherwise
Mean= 1/p = 1/0.42 = 2.380
Variance = 1-p/ p2
Variance = 1-0.42 /0.422
Variance = 3.287
2. If the probability of breaking the pot in the pool is 0.4, find the number of brakes before success and the corresponding variance and standard deviation.
Solution:
Here,
X ∼ geo(0.4)
Hence,
e(x) = 1/0.4 = 2.5
We can expect to pot off the break after 2.5 goes.
Var(x) = 0.6/0.4²
= 3.75
Hence, standard deviation ( σ) = 1.94
Quiz Time
1. Which of the following statements is true about geometric distribution?
Geometric distribution is approximately skewed right
Geometric distribution is approximately symmetric
Geometric distribution is approximately skewed left
The shape can be either symmetric or skewed.
2. Which of the following statements is not true about the geometric distribution?
Trails should be fixed
The probability of success should be 0.5
Events should be independent
Their distributions are approximately symmetric.
3. What will be the variance of geometric distribution having parameter p = 0.72?
54%
76%
13%
69%
FAQs on Geometric Distribution
1. What is a geometric distribution in statistics?
A geometric distribution is a type of discrete probability distribution that models the number of trials required to achieve the first success in a sequence of independent Bernoulli trials. Each trial has only two outcomes (success or failure), and the probability of success, denoted by p, remains constant for every trial. For example, it can be used to calculate the probability that the first head appears on the third coin toss.
2. How does a geometric distribution differ from a binomial distribution?
The key difference lies in what they measure. A geometric distribution focuses on the 'waiting time' for an event, while a binomial distribution focuses on the frequency of an event.
- Objective: A geometric distribution calculates the probability of achieving the first success on a specific trial (e.g., the 5th trial). A binomial distribution calculates the probability of a specific number of successes within a fixed number of trials (e.g., 3 successes in 10 trials).
- Number of Trials: The number of trials is variable and potentially infinite in a geometric distribution, as we wait for the first success. In a binomial distribution, the number of trials (n) is fixed from the start.
3. What is the formula for the probability mass function (PMF) of a geometric distribution?
The probability mass function (PMF) for a geometric distribution, which calculates the probability of the first success occurring on the k-th trial, is given by the formula:
P(X = k) = (1-p)k-1p
Where:
- k is the number of the trial on which the first success occurs (k = 1, 2, 3, ...).
- p is the probability of success on a single trial.
- (1-p) is the probability of failure on a single trial.
This formula represents the probability of having (k-1) consecutive failures followed by one success.
4. What are the key properties, including the mean and variance, of a geometric distribution?
A geometric distribution has several important properties that describe its central tendency and spread:
- Mean or Expected Value (E[X]): The average number of trials needed to get the first success is 1/p. If the probability of success is 0.2, you would expect to wait, on average, 1/0.2 = 5 trials.
- Variance (Var(X)): The variance, which measures the spread of the distribution, is (1-p)/p2. A higher variance means the number of trials to get a success is more unpredictable.
- Mode: The most likely number of trials for the first success is always 1, as this has the highest probability, P(X=1) = p.
- Memoryless Property: This unique property states that the probability of future outcomes does not depend on past failures.
5. What are some real-world examples of the geometric distribution?
The geometric distribution is used to model scenarios focused on waiting for an initial success. Some practical examples include:
- Quality Control: Testing products from an assembly line until the first defective item is found.
- Sales and Marketing: Making cold calls until the first successful sale is achieved.
- Sports Analytics: Calculating the probability that a basketball player makes their first free throw on their third attempt.
- Genetics: Determining the number of organisms to be sampled before finding one with a specific recessive trait.
6. Why is it called the 'geometric' distribution?
The name 'geometric' comes from its direct relationship with a geometric progression (or geometric series). The sequence of probabilities for the first success occurring on the 1st, 2nd, 3rd, and subsequent trials is p, (1-p)p, (1-p)2p, ... This sequence is a geometric progression where each term is found by multiplying the previous one by a common ratio of (1-p). The sum of all these probabilities forms an infinite geometric series that converges to 1, a fundamental requirement for any probability distribution.
7. What is the 'memoryless property' of the geometric distribution and why is it important?
The memoryless property is a fundamental concept of the geometric distribution. It states that the probability of waiting for a certain number of additional trials for a success is not affected by the number of failures already observed. Formally, P(X > m+n | X > m) = P(X > n).
In simpler terms, if you have been flipping a coin and have not yet seen a head after 10 flips, the probability of needing at least 3 more flips to get a head is the same as if you were starting from scratch. The process 'forgets' the past failures. This property is crucial for modelling systems where the probability of an event remains constant over time, regardless of history, such as the waiting time for a radioactive particle to decay.
8. Can a geometric distribution have a Probability Density Function (PDF)?
No, a geometric distribution cannot have a Probability Density Function (PDF). The geometric distribution is a discrete probability distribution because its variable (the number of trials) can only take distinct integer values (e.g., 1, 2, 3, ...). Its probabilities are defined by a Probability Mass Function (PMF), which assigns a specific probability to each integer outcome.
A PDF is used exclusively for continuous probability distributions, where the variable can take any value within a given range (e.g., height or temperature).
9. How is the Cumulative Distribution Function (CDF) for a geometric distribution calculated and interpreted?
The Cumulative Distribution Function (CDF) of a geometric distribution calculates the probability that the first success occurs on or before a specific trial, k. It is interpreted as the cumulative probability of success within a certain number of attempts.
The formula for the CDF is:
F(k) = P(X ≤ k) = 1 - (1-p)k
For example, if the probability of winning a game on any attempt is p=0.1, the CDF for k=5 would tell you the total probability of winning for the first time on the 1st, 2nd, 3rd, 4th, or 5th attempt.
10. Under what specific conditions must a scenario meet to be modelled by a geometric distribution?
For a real-world scenario to be accurately modelled by a geometric distribution, it must satisfy four key conditions derived from the Bernoulli process:
- Two Outcomes: Each trial must result in one of only two possible outcomes, which are labelled as 'success' or 'failure'.
- Independent Trials: The outcome of any trial must be independent of the outcomes of all previous trials.
- Constant Probability of Success: The probability of success (p) must remain the same for every single trial.
- Variable of Interest: The primary goal must be to determine the number of trials required to observe the very first success.
If any of these conditions are not met, a geometric distribution may not be the appropriate model.
