Binomial Distribution: Some Notes | Www.ilearnmath.net

The Binomial Distribution| AP Statistics | SHUBLEKA    

Binomial Setting 1. Each observation falls into one of just two categories, which for convenience, we call “success” and “failure”. 2. There is a fixed number n of observations. 3. The n observations are all independent. 4. The probability of success, call it p, is the same for each observation. For data produced in a binomial setting, let X = number of successes. X is called a binomial random variable and its probability distribution is called binomial distribution. Binomial Distribution The distribution of the count X of successes in the binomial setting is the binomial distribution with parameters n and p. The possible values of X are whole numbers between 0 and n. As an abbreviation, write X ~ B(n, p). Sampling distribution of count: Choose an SRS of size n from a population with proportion p of successes. When the population is much larger than the sample, the count X of successes in the sample has approximately the binomial distribution with parameters n and p. Binomial Coefficient

⎛n⎞ n! = number of ways of arranging k successes among n observations ⎜ ⎟= ⎝ k ⎠ k !( n − k ) ! Binomial Probability

⎛n⎞ P( X = k ) = ⎜ ⎟ p k (1 − p)n −k ⎝k ⎠ Example: n=10, p=0.1 Find the probability that no more than one switch fails.

P ( X ≤ 1) = P ( X = 0) + P ( X = 1) = ⎛10 ⎞ ⎛10 ⎞ = ⎜ ⎟ 0.10 0.910 + ⎜ ⎟ 0.110.99 = 0.7361 ⎝0 ⎠ ⎝1 ⎠ TI-83: binompdf 2nd, (DISTR) command: binompdf(10, 0.1, 0) + binompdf(10, 0.1, 1) = 0.361 TI-89: tistat.binomPdf(10, 0.1, 0) Cdf = cumulative distribution function

The Binomial Distribution| AP Statistics | SHUBLEKA    

Binomial Mean and Standard Deviation

μ = np σ = np (1 − p )

Example: Bad switches.

Normal Approximation for the Binomial When n is large, X ~ N (np, np (1 − p ) Conditions: np ≥ 10; n(1 − p) ≥ 10 Applet: Normal approximation to the Binomial Proportion

pˆ = X / n μ pˆ = p

σ pˆ =

p(1 − p ) n

Next: Exploring Binomial Distributions with the graphing calculator