AP Statistics
|Binomial Distributions |Objectives: |
| |-identify a random variable as binomial by verifying four conditions |
| |-use a calculator or the formula to determine binomial probabilities |
| |-calculate cumulative distribution functions for binomial r.v. |
|Binomial Setting |-calculate means and standard deviations of binomial r.v. |
| |-use a normal approximation to the binomial distribution to compute probabilities |
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|Conditions |A binomial setting arises when we perform several independent trials of the same chance process and record the number of |
| |times that a particular outcome occurs. |
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| |1. Binary? The possible outcomes of each trial cab ne classified as “success” or “failure”. |
| |2. Independent? Trials must be independent; that is, kmowing the result of one trial must not have any effect on the result|
|Binomial random variable |of any other trial. |
| |3. Number? There is a fixed number n of trials. |
|Binomial distribution |4. Success? The probability of success, we’ll call p, is the same for each trial. |
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| |If data are produced in a binomial setting, then the random variable X = number of successes is called the binomial random |
|Binomial Probability Formula |variable. |
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| |The probability distribution of X is called a binomial distribution with parameters n and p. The possible values of X are |
| |the whole numbers from 0 to n. As an abbreviation, X is B (n,p). |
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| |n = number of trials |
| |p = probability of success |
|Mean and S.D.of a Binomial Random |1 – p = probability of failure |
|Variable |X = number of successes in n trials |
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| |P(X = k) = [pic], where [pic] |
| |for k = 0,1,2,…,n. |
|Finding Binomial Probabilities Using | |
|Calculator: |Mean: μ = np Standard Deviation: σ = [pic] |
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| |pdf – given a discrete r.v. X, the probability distribution function (pdf) assigns a probability to each value of X. |
| |binompdf (n,p,X) |
|Assignment 8.1, #8.1C, 8.3C, 8.6, | |
|8.8, 8.9C, 8.10, 8.11C, 8.13, |cdf – Given a r.v. X, the cumulative distribution function (cdf) of X calculates the sum of the probabilities for 0,1,2,…, |
|8.19abcC, 8.30, 8.32, 8.34 |up to the value X. That is it calculates the probability obtaining the probability of obtaining at most X successes in n |
| |trials. |
| |binomcdf (n,p,X) |
|8.2 The Geometric Distribution | |
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|Geometric Setting | |
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|Conditions | |
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|Geometric random variable | |
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|Geometric distribution | |
| |Objectives: |
| |-identify a random variable as geometric by verifying four conditions |
|Geometric Probability Formula |-use a calculator or the formula to determine geometric probabilities |
| |-calculate cumulative distribution functions for geometric r.v. |
| |-calculate means and standard deviations of geometric r.v. |
|Mean and S.D.of a Binomial Random | |
|Variable |A geometric setting arises when we perform independent trials of the same chance process and record the number of trials |
| |until a particular outcome occurs. |
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|Finding Geometric Probabilities on |1. Binary? The possible outcomes of each trial cab ne classified as “success” or “failure”. |
|Calculator |2. Independent? Trials must be independent; that is, knowing the result of one trial must not have any effect on the result|
| |of any other trial. |
| |3.Trials? The goal is to count the number of trials until the first success occurs. |
| |4. Success? The probability of success, we’ll call p, is the same for each trial. |
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| |The number of trials Y that it takes to get a success in a geometric setting is a geometric random variable. |
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| |The probability distribution of Y is a geometric distribution with parameters p, the probability of a success on any trial. |
| |The possible values of Y are 1,2,3,…. |
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| |[pic] n = nth trial |
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|Assignment 8.2, #8.37C, 8.45C, 8.46, | |
|8.47ab, 8.49, 8.51 |Mean = ( = 1/p Standard deviation = ( = [pic] |
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| |P(X = n) use geometpdf(p,x) |
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| |P(X < n) use geometcdf(p,x) |
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