Glossary
Binomial Distribution
A probability distribution that counts the number of successes in a fixed number of independent Bernoulli trials.
Example:
Counting how many times you correctly guess on 10 multiple-choice questions would involve a binomial distribution.
Discrete
A characteristic of a random variable indicating it can only take on a countable number of distinct values, typically whole numbers.
Example:
The number of flips it takes to get the first head is a discrete variable, as you can't have 2.5 flips.
First Success Focus
The defining characteristic of a geometric distribution, where the variable of interest is the trial number on which the very first success occurs.
Example:
If you're waiting for the first time a specific song plays on the radio, you have a first success focus.
Geometric Random Variable
A discrete random variable that counts the number of independent trials needed to obtain the first success.
Example:
The number of times a basketball player shoots until they make their first basket is a geometric random variable.
Independent Trials
A condition where the outcome of one trial does not influence the outcome of any subsequent trials.
Example:
When drawing cards with replacement, each draw is an independent trial because the deck's composition resets.
Mean (Expected Value)
The average number of trials one would expect to need to achieve the first success in a geometric distribution, calculated as 1/p.
Example:
If the probability of hitting a target is 0.2, the mean (expected value) number of shots until the first hit is 1/0.2 = 5 shots.
Probability Mass Function (PMF)
A function that gives the probability that a discrete random variable is exactly equal to some specific value.
Example:
Using the Probability Mass Function, you can calculate the exact chance that the first person to correctly answer a trivia question is the 7th person asked.
Skewed Right
A characteristic shape of a distribution where the tail extends further to the right, indicating that most of the data points are concentrated on the lower end.
Example:
A histogram showing the number of attempts until a first success will typically be skewed right, as it's more likely to succeed early.
Standard Deviation
A measure of the typical variability or spread of the number of trials around the mean in a geometric distribution.
Example:
A high standard deviation for the number of attempts to get a first success means the actual number of attempts can vary greatly from the average.
Two Outcomes
A condition for Bernoulli trials where each trial can only result in one of two possibilities: success or failure.
Example:
Deciding if a customer makes a purchase or not represents the two outcomes needed for a geometric setting.
geometricCDF
A calculator function used to find the cumulative probability that the first success in a geometric distribution occurs on or before a specific trial 'k'.
Example:
If you want to know the probability that you find a parking spot within 3 attempts, you'd use geometricCDF.
geometricPDF
A calculator function used to find the probability that the first success in a geometric distribution occurs on a specific trial 'k'.
Example:
To find the probability that the first time you roll a 'snake eyes' (double ones) is on your 5th roll, you would use geometricPDF.