AP Statistics Ch 8: Binomial and geometric distributions
Part 1: Binomial distributions
- it is vital to know when a situation is a binomial setting and when it is a geometric setting (covered in part 2)
- what makes a setting a binomial setting
- outcomes can only be success and failure
- there is a fixed number n of observations (trial)
- the observations are independent, and knowing one observation won’t help you know the next observation
- probability of success is the same for every trial
- if the situation is a binomial setting
- binomial random variable: the number of successes of the random variable, described by X
- binomial distribution: probability distribution of X, or the distribution of (number of successes)/(number of trials)
- has parameters of n (or number of trials) and p (or the probability of success for each trial)
- X can be any number from 0 to n
- can be abbreviated B(n, p)
- is a discrete probability distribution
- important to know when the binomial distributions apply
- Binomial distribution is sampling
- used to find out about the probability of success p in a population
- even though using simple random sampling can result in making the trials dependent, if the population is much larger than the sample, then the count of successes p in an SRS with n trials is about the same as the p obtained in the binomial distribution with the same n number of trials
- in other words, in this situation, the p obtained with an SRS of size n is about the same as B(n, p) with the same size n
- the equations for finding probability for a binomial distribution, called binomial probability
- nCk (pk)(1-p)n-k
- k = the number of success, n = the number of trials
- nCk is the binomial coefficient, which is the number of ways of arranging the successes among the observations
- can use the calculator to calculate, or use (n!)/(k!(n-k)!)
- using pdf
- called the probability distribution function
- given an X number of successes, pdf can tell you what is the probability that X number of successes will occur
- using cdf
- we can use this if we want to calculate probabilities for more than one X
- for example, if we calculate P(x< or equal to 3), we can use the cdf function
- called the cumulative distribution function
- if you want to calculate the probability of P(x greater or equal than #), then subtract P(x equal or less than #-1) from 1
- the idea here is that cdf only calculates the probability for x is equal or less than ___, not the probability for x is equal or greater than ___
- mean and standard deviation of binomial distribution
- in general, mean should be np, where n is the total number of trials and p is the probability of success in each trial
- standard deviation is (np(1-p))0.5
- IMPORTANT! These formulas for mean and standard deviation can only be used for binomial distributions!
- Normal approximations for binomial distributions
- as the number of trials increase, the binomial distribution will get close to a Normal distribution, so we can then use Normal probability calculations to find probabilities
- notation will be N(np, (np(1-p))0.5
- we can use this when np is greater or equal to 10 and when n(1-p) is greater or equal to 10
- accuracy improves as n increases, and is most accurate when p is equal to 0.5
Part 2: Geometric Distributions
- unlike binomial distributions, where the number of trials is known and fixed, we want to know when (what trial #) we will get our first success in geometric distributions
- takes place in geometric settings
- requirements of geometric settings
- only two categories: fail and success
- observations are independent
- probability p for success is the same for every observation
- we want to know the number of trials required to obtain the first success
- so X = number of trials until ____occurs
- equation for geometric distributions
- the probability that the first success occurs on the nth trial = (1-p)n-1(p)
- simplifies to P(X=n) = (1-p)n-1(p)
- n = trial number
- the probability that the first success will take more than n trials to occur = (1-p)n
- distribution table for geometric series never ends
- properties of geometric random variable
- the expected value is just another way to say the mean; it is the expected number of trials required to get the first success
- mean = 1/p
- variance = (1-p)/(p2), and the standard deviation is just the square root of that
Websites I found:
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