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  2. AP Statistics
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What is the formula for the PMF of a geometric distribution?

P(Y=k)=(1−p)k−1∗pP(Y = k) = (1-p)^{k-1} * pP(Y=k)=(1−p)k−1∗p

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What is the formula for the PMF of a geometric distribution?

P(Y=k)=(1−p)k−1∗pP(Y = k) = (1-p)^{k-1} * pP(Y=k)=(1−p)k−1∗p

What is the formula for the mean (expected value) of a geometric distribution?

E(Y)=1pE(Y) = \frac{1}{p}E(Y)=p1​

What is the formula for the standard deviation of a geometric distribution?

σY=1−pp2σ_Y = \sqrt{\frac{1-p}{p^2}}σY​=p21−p​​

Given p = 0.4, what is E(Y)?

E(Y) = 1/0.4 = 2.5

Given p = 0.5, what is the standard deviation?

σY=1−0.50.52=1σ_Y = \sqrt{\frac{1-0.5}{0.5^2}} = 1σY​=0.521−0.5​​=1

What is a geometric random variable?

Counts the number of trials needed to get the first success in a series of independent trials.

Define 'success' in a geometric distribution.

One of two possible outcomes in each independent trial, with a fixed probability p.

Define 'failure' in a geometric distribution.

The other possible outcome (besides success) in each independent trial, with a probability of 1-p.

What is the Probability Mass Function (PMF)?

The probability of the first success occurring on the kth trial.

What does geometricCDF(p, k) calculate?

Probability of the first success on or before trial k (cumulative probability).

What are the differences between geometric and binomial distributions?

Geometric: Counts trials to first success | Binomial: Counts successes in a fixed number of trials.

Compare the focus of geometric and binomial distributions.

Geometric: Focus on the trial number of the first success. | Binomial: Focus on the number of successes.

What are the conditions for Binomial distribution?

Binary outcome, Independent trials, Number of trials fixed, Success probability fixed.

What are the conditions for Geometric distribution?

Go until first success, Independent trials, First success, Trials not fixed.

Compare the shape of Binomial and Geometric Distributions.

Geometric: Skewed Right | Binomial: Can be skewed or approximately normal depending on p and n.