Geometric Distribution formula derivation properties and solved problems
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 in Probability Explained Clearly
1. What is a geometric distribution?
A geometric distribution is a discrete probability distribution that models the number of trials needed to get the first success in repeated independent Bernoulli trials with constant probability of success p.
- Each trial has only two outcomes: success or failure.
- The probability of success remains constant in every trial.
- It answers questions like: “How many trials until the first success?”
2. What is the formula for the geometric distribution?
The probability mass function of a geometric distribution is P(X = k) = (1 − p)k−1 p, where k = 1, 2, 3, ....
- p = probability of success
- 1 − p = probability of failure
- k = trial on which the first success occurs
3. What is the mean of a geometric distribution?
The mean (expected value) of a geometric distribution is E(X) = 1/p.
- If p = 0.25, then E(X) = 1/0.25 = 4.
- This means, on average, it takes 4 trials to get the first success.
4. What is the variance of a geometric distribution?
The variance of a geometric distribution is Var(X) = (1 − p)/p2.
- If p = 0.5, then Var(X) = (0.5)/(0.5)2 = 2.
5. How do you calculate probability using a geometric distribution?
To calculate probability in a geometric distribution, use the formula P(X = k) = (1 − p)k−1 p and substitute the given values.
- Example: If p = 0.3, find probability of first success on 3rd trial.
- P(X = 3) = (0.7)2 × 0.3 = 0.49 × 0.3 = 0.147.
6. What is the memoryless property of the geometric distribution?
The geometric distribution has the memoryless property, meaning P(X > s + t | X > s) = P(X > t).
- The probability of waiting additional trials does not depend on how many trials have already occurred.
- Past failures do not change future probabilities.
7. What is the difference between geometric and binomial distribution?
The key difference is that the geometric distribution counts trials until the first success, while the binomial distribution counts the number of successes in a fixed number of trials.
- Geometric: Number of trials is random, first success is fixed.
- Binomial: Number of trials is fixed, successes are counted.
- Geometric uses P(X = k) = (1 − p)k−1 p.
- Binomial uses P(X = k) = C(n,k)pk(1−p)n−k.
8. Can you give an example of a geometric distribution?
An example of a geometric distribution is flipping a fair coin until the first head appears.
- Probability of head p = 0.5.
- Probability first head occurs on 4th flip:
- P(X = 4) = (0.5)3 × 0.5 = 0.0625.
9. When should you use a geometric distribution?
You should use a geometric distribution when you are counting the number of independent trials until the first success with constant probability p.
- Trials are independent.
- Only two outcomes: success or failure.
- Probability of success stays constant.
10. What are the conditions for a geometric distribution?
A geometric distribution applies when specific conditions are satisfied:
- Each trial is a Bernoulli trial (two outcomes).
- Trials are independent.
- Probability of success p is constant.
- The random variable counts trials until the first success.
