Continuous Probability Distributions. Discrete vs. Continuous Discrete ▫A random variable (RV)...
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![Page 1: Continuous Probability Distributions. Discrete vs. Continuous Discrete ▫A random variable (RV) that can take only certain values along an interval:](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697bf791a28abf838c8240f/html5/thumbnails/1.jpg)
Continuous Probability Distributions
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Discrete vs. Continuous
•Discrete▫A random variable (RV) that can take only
certain values along an interval: Cars passing by a point Results of coin toss Students taking a class
•Continuous▫An RV that can take on any value at any
point along an interval.
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Continuous Probability Distributions
•Discrete: For any random variable X: P(X=x)
•Continuous: ▫The probability that a continuous random
variable will assume a specific value is zero▫Therefore, a continuous random variable
cannot be expressed in tabular form.▫An equation or formula is used to describe
a continuous random variable. This is called a probability density function (pdf)
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Limits (kind of)
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•The random variable is a function of X▫y = f(x)
•The value of f(x) is greater than or equal to zero for all values of x.
•The total area under the curve always equals one.
Probability Density Functions
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Continuous Probability Distributions
Let’s assume that a train arrives at the station precisely every 30 minutes.
If passengers arrive at the station at random intervals, what is the probability…?
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Continuous Distributions
•Normal distribution•Standard normal distribution•Exponential distribution•Chi-square distribution•F distribution
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Normal Distribution
•Carl Friedrich Gauss
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Normal Distribution
•Many natural and economic phenomena are normally distributed
•The normal can approximate other distributions, including the binomial
•Sample proportions are normally distributed when taken from a population of any distribution
•Normal is a family of distributions▫Mean, median, and mode all at the same
position▫Curve is symmetric▫Curve is asymptotic
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pdf for the Normal
2σ
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Empirical Rule
±1σ = 68%
±2σ = 95%
±3σ = 99.7%
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Example – Empirical Rule
•Scores on a standardized test are normalized with a mean of 500
•Assume a normal distribution with a standard deviation of 100
•What is the probability a randomly selected student’s score will be:▫More than 600▫Between 300 and 500▫Less than 400▫Between 400 and 700
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Standard Normal Distribution
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Standardizing Individual Data Values
• The standardized z-score is how far above or below the individual value is compared to the population mean in units of standard deviation.▫“How far above or below”= data value – mean▫“In units of standard deviation”= divide by s
© 2008 Thomson South-Western
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ExampleThe average hotel check-in time is 12 minutes. Mary just left the cab that brought her to her hotel. Assuming a normal distribution with a standard deviation of 2.0 minutes, what is the probability that the time required for Mary and her bags to get to the room will be:
a) greater than 14 minutes?b) less than 8.5 minutes?c) between 10.5 and 14.0 minutes?
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Example - CDF
•An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assuming that bulb life is normally distributed, what is the probability that an Acme light bulb will last at most 365 days?
•http://davidmlane.com/hyperstat/z_table.html
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More Practice•The average charitable contribution
among people making $60,000 - $75,000 is $1935.
•Assume donations are normally distributed
•Assume a standard deviation of $400.▫What’s the probability that a randomly
selected person in this category made charitable contributions of at least $1600?
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Normal Approximation of the Binomial•Continuity correction
▫Add or subtract .5 to correct for the gaps•Useable when:
▫nπ and n(1-π) are both >+5
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Practice
•An expert claims there is no difference between the taste of 2 soft drinks.
•In a taste test involving 200 people, 55% of the testers preferred soft drink A.
• If the expert was correct, what’s the probability that 110 or more of the testers would prefer soft drink A?