10.8 - Probability Distributions and Binomial Distributions

These flashcards cover key vocabulary and formulas related to probability distributions and binomial distributions, including definitions, formulas, and important concepts.

Probability Distribution

A function or rule that assigns probabilities to each possible outcome of a random variable.

Binomial Distribution

Calculates the probability of an event occurring a specified number of times in a fixed number of trials, with two possible outcomes (success/failure).

Expected Value (Mean)

The average value of a random variable, calculated as the sum of all possible values, each multiplied by the probability of its occurrence.

Variance

The measure of the dispersion of a set of values; in probability, it quantifies how far the values deviate from the expected value.

Standard Deviation

The square root of variance, representing the average distance of each data point from the mean.

Conditions for Binomial Distribution

1. Each trial is binary (success/failure).

2. The trials are independent.

3. There is a fixed number of trials.

4. The probability of success is constant.

Notation for Binomial Distribution: n, p, X

n = number of trials

p = probability of success

X = desired number of successes.

Binomial Probability Formula

For exactly X successes in n trials: $P(X) = nC_x imes p^x imes (1 - p)^{n - x}$.

Complement Rule for Binomial Distribution

To find the probability of at least k successes, calculate: $P(X \geq k) = 1 - P(X \leq k-1)$.

Mean of Binomial Distribution

The mean (expected value) is given by the formula: $\mu = n p$.

Variance of Binomial Distribution

The variance is given by the formula: $\sigma^2 = n p (1 - p)$.

Standard Deviation of Binomial Distribution

The standard deviation is given by the formula: $\sigma = \sqrt{n p (1 - p)}$.

Normal Distribution

A continuous probability distribution characterized by a symmetric, bell-shaped curve, defined by its mean (μ) and standard deviation (σ).