Geometric Distribution

Geometric distribution

• A discrete probability distribution that models the number of independent Bernoulli trials needed to achieve the first success

• Used in reliable testing to predict how many cycles a component will run before failure, given a constant failure probability.

Bernoulli Trial

• A random experiment with two possible outcomes: success or failure, where the probability of success remains constant across trials

• testing whether an LED works when powered on with “working” as success and “not working” as failure.

Probability of success (p)

• The probability that any given trial results in a success.

• If a light bulb has a 0.98 probability of passing a quality check, then p = 0.98

Probability of Failure (1 - p)

• The probability that a given trial results in a failure.

• For a 2% defect rate in manufactured chips, 1 - p = 0.02

Random Variable (X) in Geometric Distribution

• Represents the number of trials until the first success occurs.

• The number of times a software engineer runs a simulation before it produces the correct output.

Probability Mass Function (PMF)

• P(X = x) = ((1 - p)^(x - 1))p

• Gives the probability that the first success occurs on the x-th trial.

• If the chance of a robot’s sensor detecting a signal on each scan is 0.3, then the probability that detection first happens on the 4th scan is (0.7)³(0.3)

Cumulative Distribution Function (CDF)

• P(X ≤ x) = 1 - (1 - p)^x

• Gives the probability that the first success happens within x trials.

• In testing 10 circuits this gives the probability that at least one passes before the 10th test.

Expected Value (Mean)

• E(X) = 1/p

• Represents the average number of trials until the first success.

• If a machine passes 20% of the time, it will take on average 5 inspections to find one that passes.

Variance of Geometric Distribution

• V(X) = (1 - p)/(p²)

• Indicates how spread out the number of trials before the first success is

• Engineers use this to estimate how inconsistent a production line’s “first pass yield” is

Standard Deviation

• σ = ((1 - p)/(p²))^1/2

• Measures the typical deviation from the mean number of trials before success

• If a test system has p = 0.25,

• Then σ = ((0.75)/(0.25)²)^1/2 = 3.46

• Showing relatively high variability

Memoryless property

• P(X > s + t|X > s) = P(X > t)

• The distribution’s future probabilities are unaffected by past failures

• In communication systems, the probability of receiving the first correct bit after t more transmissions doesn’t depend on previous transmission failures.

Support of the Geometric Distribution

• The possible values of X are 1, 2, 3, …; X cannot be 0 since at least one trial is needed

• Counting the number of component tests before one works; you must test at least one.

Expected Number of Failures before First Success

• E(X - 1) = (1 - p)/p

• Represents how many failures occur on average before success.

• If 10% of circuits pass, you expect an average of 9 defective circuits before finding a working one.

Reliability Engineering Interpretation

• The geometric distribution models the number of operational cycles before the first failure when the failure rate is constant

• In a drone’s rotor testing, if each flight had 0.99 success rate, the distribution predicts how many flights occur before a malfunction.

Quality Control Example

• Used to estimate how many products will be tested before finding the first defective unit

• If there’s a 1% defect rate, the probability that the first defective unit appears on the 50th item is ((0.99)^49)(0.01)

Software Reliability Example

• The geometric model can represent how many attempts are needed before the first successful execution of a program after debugging

• If each recompile has 0.2 chance of fixing a bug, then the expected number of attempts is 5

Network Communication Example

• Describes how many packet transmissions are needed before the first one is successfully received.

• If each packet has a 0.95 chance of correct transmission, the average number of tries before success is 1/0.95 = 1.05

Industrial Process Example

• Models how many attempts it takes for a machine to produce a perfect part after recalibration.

• With p = 0.85, the probability that success occurs on the 3rd attempt is (0.15)²(0.85).

Binomial Distribution v/s Geometric Distribution

• In a binomial distribution, the number of successes is fixed

• In a geometric distribution, the number of trials is variable until the first success

• The binomial distribution might model “how many working sensors in 10 tests,”

• The geometric distribution models “How many tests until the first sensor”

Geometric distribution and Negative Binomial Distribution

• The geometric distribution is a special case of the negative binomial where the required successes r = 1.

• Negative Binomial distribution might represent the number of tests until 3 working circuits

• while geometric distribution represents until the first.