Wait for a success
In independent trials with a fixed success probability, the number of failures before the first success follows the geometric distribution under the failures-count convention.
Probability & statistics / English
How to Learn Probability Distributions
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Connect probability distributions through the stories that generate counts and waiting times.
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1:30Wait for a success3:00Change what you count4:30Continuous waiting timesLearning script
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In independent trials with a fixed success probability, the number of failures before the first success follows the geometric distribution under the failures-count convention.
Counting successes in a fixed block of trials gives a binomial variable. Counting failures before a fixed number of successes gives a negative binomial variable.
A suitable rare-event limit links these stories to Poisson counts and exponential waiting times. Sums of independent exponential waits with the same rate give gamma waiting times.
Knowledge cards
Change what you count
Counting successes in a fixed block of trials gives a binomial variable. Counting failures before a fixed number of successes gives a negative binomial variable.
Continuous waiting times
A suitable rare-event limit links these stories to Poisson counts and exponential waiting times. Sums of independent exponential waits with the same rate give gamma waiting times.
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- Distributions ExplanationAt 1:30
Why this connection?
In independent trials with a fixed success probability, the number of failures before the first success follows the geometric distribution under the failures-count convention. Counting successes in a fixed block of trials gives a binomial variable. Counting failures before a fixed number of successes gives a negative binomial variable. A suitable rare-event limit links these stories to Poisson counts and exponential waiting times. Sums of independent exponential waits with the same rate give gamma waiting times.
Questions this video answers
Meet the concept↗
A suitable rare-event limit links these discrete stories to Poisson counts and exponential waiting times.
Conditions: The context is a rare-event limit.; The variables represent counts and waiting times.
3:00 – 4:30Watch this moment ↗
Meet the concept↗
Sums of independent exponential waits with the same rate give gamma waiting times.
Conditions: The exponential waits are independent.; All exponential waits share the same rate.
4:30 – 6:00Watch this moment ↗
Meet the concept↗
The geometric distribution describes the number of failures before the first success, provided the trials are independent and the success probability remains fixed.
Conditions: Trials are independent.; Success probability is fixed.; The convention counts the number of failures before the first success.
0:00 – 1:30Watch this moment ↗
Meet the concept↗
Counting successes in a fixed block of trials gives a binomial variable, while counting failures before a fixed number of successes gives a negative binomial variable.
Conditions: Trials are independent with a fixed success probability.; The first scenario involves a fixed total number of trials.; The second scenario involves waiting for a fixed number of successes.
1:30 – 3:00Watch this moment ↗
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