The Geometric Distribution - AP Statistics

$P(X \leq 5) = 1 - (0.8)^5$. CDF calculation for success within 5 trials.

$P(X = k) = (1-p)^{k-1}p$. Standard PMF where $k$ is trial number and $p$ is success probability.

The number of trials until the first success. Counts trials needed to achieve one success in independent trials.

$E(X) = \frac{1}{p}$. Higher success probability means fewer expected trials needed.

Trials are independent with constant probability $p$ of success. Each trial must have same success chance and be unaffected by others.

$Var(X) = \frac{1-p}{p^2}$. Standard variance formula derived from geometric distribution theory.

The geometric distribution. Specifically models waiting time until first success occurs.

$P(X = 4) = (0.7)^3 \times 0.3$. Three failures $(1-0.3)^3$ then success $0.3$ on trial 4.

Positively skewed. Most probability mass at $X=1$, decreasing for higher values.

As $p$ increases, the mean decreases. Higher $p$ means success comes sooner on average.

$P(X > 3) = (1-0.2)^3$. Probability of needing more than 3 trials equals $(0.8)^3$.

Mode = 1. First trial always has highest probability of success.

$E(X) = \frac{1}{0.5} = 2$. Expected trials equals reciprocal of success probability.

$Var(X) = \frac{1-0.4}{0.4^2}$. Uses standard variance formula $\frac{1-p}{p^2}$ with $p=0.4$.

$P(X = k) = (1-p)^{k-1}p$. Standard PMF where $k$ is trial number and $p$ is success probability.

The number of trials until the first success. Counts trials needed to achieve one success in independent trials.

$P(X \leq k) = 1 - (1-p)^k$. CDF gives probability of success by trial $k$.

$P(X = 4) = (0.7)^3 \times 0.3$. Three failures $(1-0.3)^3$ then success $0.3$ on trial 4.

$P(X \leq 3) = 1 - (0.5)^3$. CDF formula gives probability of success by trial 3.

Trials are independent. Each trial outcome doesn't affect subsequent trial probabilities.

$E(X) = \frac{1}{p}$. Higher success probability means fewer expected trials needed.

Trials are independent with constant probability $p$ of success. Each trial must have same success chance and be unaffected by others.

$Var(X) = \frac{1-p}{p^2}$. Standard variance formula derived from geometric distribution theory.

$P(X > 3) = (1-0.2)^3$. Probability of needing more than 3 trials equals $(0.8)^3$.

Mode = 1. First trial always has highest probability of success.

$E(X) = \frac{1}{0.5} = 2$. Expected trials equals reciprocal of success probability.

$Var(X) = \frac{1-0.4}{0.4^2}$. Uses standard variance formula $\frac{1-p}{p^2}$ with $p=0.4$.

Discrete random variable. Counts whole number trials, not continuous values.

$P(X \leq 4) = 1 - (0.7)^4$. Uses CDF formula with success probability $p=0.3$.