Chapter 7 Random Variables 7.1: Discrete and Continuous Random Variables.
Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete...
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Transcript of Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete...
![Page 1: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/1.jpg)
Statistics
Chapter 6 / 7 Review
![Page 2: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/2.jpg)
Random Variables and Their Probability Distributions
• Discrete random variables – can take on only a countable or finite number of values.
• Continuous random variables – can take on countless values in an interval on the real line
• Probability distributions of random variables – An assignment of probabilities to the specific values or a range of values for a random variable.
![Page 3: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/3.jpg)
Discrete Probability Distributions
1) Each value of the random variable has an assigned probability.
2) The sum of all the assigned probabilities must equal 1.
![Page 4: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/4.jpg)
Means and Standard Deviations for Discrete Probability Distributions
![Page 5: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/5.jpg)
Binomial Experiments
1) There are a fixed number of trials. This is denoted by n.
2) The n trials are independent and repeated under identical conditions.
3) Each trial has two outcomes:
S = success F = failure
![Page 6: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/6.jpg)
Binomial Experiments
4) For each trial, the probability of success, p, remains the same. Thus, the probability of failure is 1 – p = q.
5) The central problem is to determine the probability of r successes out of n trials.
![Page 7: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/7.jpg)
Binomial Probability Formula
![Page 8: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/8.jpg)
Binomial Probability Formula
Find the probability of observing 6 successes in 10 trials if the probability of success is p = 0.4.
a). 0.111 b). 0.251 c). 0.0002 d). 0.022
![Page 9: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/9.jpg)
Binomial Probability Formula
Find the probability of observing 6 successes in 10 trials if the probability of success is p = 0.4.
a). 0.111 b). 0.251 c). 0.0002 d). 0.022
![Page 10: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/10.jpg)
Binomial Probabilities
• At times, we will need to calculate other probabilities:• P(r < k)• P(r ≤ k)• P(r > k)• P(r ≥ k)
where k is a specified value less than or equal to the number of trials, n.
![Page 11: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/11.jpg)
Mean and Standard Deviation of a Binomial Distribution
npq
np
![Page 12: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/12.jpg)
Critical Thinking
• Unusual values – For a binomial distribution, it is unusual for the number of successes r to be more than 2.5 standard deviations from the mean.
– This can be used as an indicator to determine whether a specified number of r out of n trials in a binomial experiment is unusual.
![Page 13: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/13.jpg)
Chapter 7
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Features of the Normal Curve
• Smooth line and symmetric around µ.
• Highest point directly above µ.
• The curve never touches the horizontal axis in either direction.
• As σ increases, the curve spreads out.
• As σ decreases, the curve becomes more peaked around µ.
• Inflection points at µ ± σ.
![Page 15: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/15.jpg)
Normal Probability
• The area under any normal curve will alwaysbe 1.
• The portion of the area under the curve within a given interval represents the probability that a measurement will lie in that interval.
![Page 16: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/16.jpg)
The Empirical Rule
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Raw Scores and z Scores
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Distribution of z-Scores
• If the original x values are normally distributed, so are the z scores of these x values.
– µ = 0– σ = 1
![Page 19: Statistics Chapter 6 / 7 Review. Random Variables and Their Probability Distributions Discrete random variables – can take on only a countable or finite.](https://reader036.fdocuments.us/reader036/viewer/2022062518/5697bf881a28abf838c897ce/html5/thumbnails/19.jpg)
Area to the Left of a Given z Value
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Area to the Right of a Given z Value
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Area Between Two z Values