Notes – Chapter 17 Binomial & Geometric Distributions

Binomial Distributions Conditions for a Binomial Distribution: 1) Only two categories (i.e. success or failure) 2) Observations are independent* 3) Probability of success is the same for each observation. 4) Fixed number of n observations

Binomial Distributions *we can use the binomial distributions in the statistical setting of selecting an SRS when the population is much larger than the sample without worrying about replacement

Binomial Distributions If the conditions for a binomial distribution are met, then a binomial distribution can be built. The binomial variable, X, is numeric and is a count of the number of successes for the given event.

Binomial Distributions Binomial notation is B(n, p) for any binomial distribution, where n is the number of trials and p is the probability

Binomial Distributions IF we are interested in a specific value of X: P(X = ?) PDF– Given a discrete random variable X, the probability distribution function assigns a probability to each value of X.

Binomial & Geometric Distributions If we are interested in a range of values for X: P(X ≤ ?) CDF – Given a random variable X, the cumulative distribution function of X calculates the sum of the probabilities for 0, 1, 2, … up to the value of X. That is, it calculates the probability of obtaining at most X successes in n trials. (See calculator use on page 393 for more info)

Binomial Distributions Binomial Coefficient The number of ways I can arranging k successes among n observations when order matters is given by the binomial coefficient for k = 0, 1, 2, …, n

Binomial Distributions Binomial Probability If X has a binomial distribution B(n, p) then…

Binomial Distributions Mean and Standard Deviation of a Binomial Distribution  = np  =  np(1 – p)

Geometric Distributions The Geometric Distribution Used when the goal is to obtain the first success. A random variable X can be defined that counts the number of trials needed to obtain that first success.

Geometric Distributions Conditions for a Geometric Distribution 1) Only two categories (i.e. success or failure) 2) Observations are independent* 3) Probability of success is the same for each observation. 4) The variable of interest is the number of trials required to obtain the first success.

Geometric Distributions *we can use the geometric distributions in the statistical setting of selecting an SRS when the population is much larger than the sample without worrying about replacement

Geometric Distributions Rules for calculating Geometric Probabilities: P(X = n) = (1 – p) n-1 p P(X > n) = (1 – p) n

Geometric Distributions The mean of a Geometric Random Variable  = 1/p

Binomial & Geometric Distributions Showing your work…. 1)Always check your conditions before using either formula. Be specific & use context. 2) Always show the numbers you used for Geometric distributions. You do not need to write the Geometric formulas.

Binomial & Geometric Distributions Showing your work…. 3) Substitute into the binomial formulas for P(x = k) only not for etc... 4) Expand and/or rewrite the probability statements for any binomials inequalities. Do not substitute into the binomial formula.

Binomial Distributions Calculator use for binomial: P(x = k) is calculated binompdf(n, p, k) So… In a binomial distribution B(20,.4), P(x = 5) is calculated binompdf(20,.4, 5)

Binomial Distributions P(x ≤ k) is calculated binomcdf(n, p, k) So… In a binomial distribution B(20,.4), P(x ≤ 5) is calculated binomcdf(20,.4, 5)

Binomial & Geometric Distributions P(x k) and P(x ≥ k) can not be calculated directly from the calculator. You must rephrase the inequality in terms of one of the previous two statements and sometimes use the complement rule.