2.7.1 Geometric Flashcards
What is a geometric distribution?
A discrete probability distribution that models the number of independent Bernoulli trials needed to get the first success.
What are the key assumptions of a geometric distribution?
- Each trial has only two outcomes (success or failure).
- The probability of success stays the same for every trial.
- The trials are independent.
- The process continues until the first success occurs.
How do we denote that a random variable ( X ) follows a geometric distribution?
( X sim ext{Geometric}(p) ), where ( p ) is the probability of success on each trial.
What is the probability that the first success happens on the ( k )-th trial?
P(X = k) = (1 - p)^{k - 1} p
This means that the first ( k - 1 ) trials must all be failures, followed by a success.
How does the geometric distribution differ from the binomial distribution?
- The binomial distribution models the number of successes in a fixed number of trials.
- The geometric distribution models the number of trials until the first success.
What is the probability that the first success happens after ( k ) trials?
P(X > k) = (1 - p)^k
What is the expected number of trials until the first success?
E[X] = rac{1}{p}
What is the variance of a geometric distribution?
ext{Var}(X) = rac{1 - p}{p^2}
What does the memoryless property of the geometric distribution state?
The probability of success does not depend on past failures.
Mathematically, for any ( k ) and ( n ):
P(X > k + n mid X > k) = P(X > n)
This means that past failures do not change future probabilities.
Example: A fair coin is flipped repeatedly until heads appears. What is the expected number of flips needed?
Since ( p = 0.5 ):
E[X] = rac{1}{0.5} = 2
Example: You roll a fair die until you get a 6. What is the probability that you roll at least 5 times?
Since ( p = 1/6 ):
P(X geq 5) = (1 - 1/6)^4 = (5/6)^4 = 0.482
What is an alternative way to define a geometric random variable?
Instead of counting the total trials until the first success, we can count the failures before the first success.
This version follows:
Y sim ext{Geometric}(p)
where ( Y = X - 1 ).
How do the PMF, mean, and variance change in this alternative definition?
- PMF: ( P(Y = k) = (1 - p)^k p ) (starts at ( k = 0 ) instead of ( k = 1 )).
- Mean: ( E[Y] = rac{1 - p}{p} ).
- Variance: ( ext{Var}(Y) = rac{1 - p}{p^2} ) (same as original version).
