Geometric Distributions

Front

What is a geometric distribution?

Back

A geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials, where each trial has the same probability of success.

Front

What is the probability of success in a geometric distribution?

Back

The probability of success (p) is the likelihood of achieving the desired outcome in a single trial.

Front

What is the formula for the probability mass function of a geometric distribution?

Back

P(X = k) = (1 - p)^(k-1) * p, where k is the number of trials until the first success.

Front

What does 'k' represent in a geometric distribution?

Back

In a geometric distribution, 'k' represents the number of trials until the first success.

Front

How do you calculate the expected value of a geometric distribution?

Back

The expected value (E[X]) of a geometric distribution is calculated as E[X] = 1/p.

Front

If a kicker has a 90% success rate, what is the expected number of attempts before a miss?

Back

E[X] = 1/(0.9) = 1.111, meaning on average, he will attempt about 1.111 kicks before missing.

Front

What is the variance of a geometric distribution?

Back

The variance (Var(X)) of a geometric distribution is given by Var(X) = (1 - p) / p^2.

Front

What is the probability of making the first success on the third trial if p = 0.75?

Back

P(X = 3) = (1 - 0.75)^(3-1) * 0.75 = 0.046875.

Front

What is the difference between a geometric random variable and a non-geometric random variable?

Back

A geometric random variable counts the number of trials until the first success, while a non-geometric random variable may not have this specific structure.

Front

In the context of fishing, when is the variable 'x' geometric?

Back

'x' is geometric if the fish are thrown back after each catch, allowing for the same probability of catching a trout on each trial.