![Page 1: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/1.jpg)
AP STATISTICS
7.3 Probability Distributions for Continuous Random
Variables
![Page 2: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/2.jpg)
7.3 Objectives:Ø Understand the definition and properties of continuous random variablesØ Be able to represent the probability distributions of continuous random variables using tables, formulae, or histograms
![Page 3: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/3.jpg)
7.3 Objectives:Ø Be able to interpret formulaic or graphic representations of probability distributions of continuous random variables.Ø Be able to calculate probabilities such as P(x < a), P(x ≥ a), and P(a ≤ x ≤ b)
![Page 4: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/4.jpg)
Let X = the weight (in pounds) of a full-term newborn child (reported to the nearest pound
x 4 5 6 7 8 9p(x) 0.26 0.31 0.21 0.13 0.06 0.03
![Page 5: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/5.jpg)
Consider the random variable:x = the weight (in pounds) of a full-term
newborn child
What type of variable is this?
![Page 6: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/6.jpg)
Consider the random variable:x = the weight (in pounds) of a full-term
newborn childSuppose that weight is reported to the nearest pound. The following probability histogram displays the distribution of weights.
The area of the rectangle centered over 7 pounds
represents the probability 6.5 < x < 7.5
![Page 7: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/7.jpg)
Consider the random variable:x = the weight (in pounds) of a full-term
newborn childSuppose that weight is reported to the nearest pound. The following probability histogram displays the distribution of weights.
What is the sum of the areas of all the rectangles?
![Page 8: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/8.jpg)
Consider the random variable:x = the weight (in pounds) of a full-term
newborn childSuppose that weight is reported to the nearest pound. The following probability histogram displays the distribution of weights.
Now suppose that weight is reported to the nearest 0.1 pound. This would be the probability histogram.
![Page 9: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/9.jpg)
Consider the random variable:x = the weight (in pounds) of a full-term
newborn childSuppose that weight is reported to the nearest pound. The following probability histogram displays the distribution of weights.
Now suppose that weight is reported to the nearest 0.1 pound. This would be the probability histogram.
Notice that the rectangles are narrower and the histogram begins to
have a smoother appearance.
![Page 10: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/10.jpg)
Consider the random variable:x = the weight (in pounds) of a full-term
newborn childSuppose that weight is reported to the nearest pound. The following probability histogram displays the distribution of weights.
Now suppose that weight is reported to the nearest 0.1 pound. This would be the probability histogram.
If weight is measured with greater and greater accuracy, the histogram
approaches a smooth curve.
![Page 11: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/11.jpg)
Consider the random variable:x = the weight (in pounds) of a full-term
newborn childSuppose that weight is reported to the nearest pound. The following probability histogram displays the distribution of weights.
Now suppose that weight is reported to the nearest 0.1 pound. This would be the probability histogram.
This is an example of a density curve.
The shaded area represents the probability 6 < x < 8.
![Page 12: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/12.jpg)
Probability Distributions for Continuous Variables
• Is specified by a curve called a density curve.
• The function that describes this curve is denoted by f(x) and is called the density function.
• The probability of observing a value in a particular interval is the area under the curve and above the given interval.
![Page 13: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/13.jpg)
Properties of continuous probability distributions
1. f(x) > 0 (the curve cannot dip belowthe horizontal axis)
2. The total area under the density curve equals one.
![Page 14: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/14.jpg)
Let X denote the amount of gravel sold (in tons) during a randomly selected week at a particular sales facility. Suppose that the density curve has a height f(x) above the value x, where
The density curve is shown in the figure:
⎩⎨⎧ ≤≤−
=otherwise0
10)1(2)(
xxxf
1
1
2
Tons
Density
![Page 15: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/15.jpg)
1 – ½(0.5)(1) = .75
Gravel problem continued . . .What is the probability that at most ½ ton of gravel is sold during a randomly selected week?
1
1
2
Tons
Density
P(x < ½) =The probability would be the
shaded area under the curve and above the interval from 0 to 0.5.
![Page 16: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/16.jpg)
1
1
2
Tons
Density
P(x = ½) =The probability would be the area
under the curve and above 0.5.
How do we find the area of a line segment?
Since a line segment has NO area, then the probability that exactly ½
ton is sold equals 0.
![Page 17: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/17.jpg)
= 1 – ½(0.5)(1) = .75
Gravel problem continued . . .What is the probability that less than ½ ton of gravel is sold during a randomly selected week?
1
1
2
Tons
Density
P(x < ½) = P(x < ½)
![Page 18: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/18.jpg)
![Page 19: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/19.jpg)
Suppose x is a continuous random variable defined as the amount of time (in minutes) taken by a clerk to process a certain type of application form. Suppose x has a probability distribution with density function:
The following is the graph of f(x), the density curve:
⎩⎨⎧ <<
=otherwise0
645.)(
xxf
0.5
4 5 6Time (in minutes)
Den
sity
![Page 20: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/20.jpg)
Application Problem Continued . . .What is the probability that it takes more than 5.5 minutes to process the application form?
0.5
4 5 6Time (in minutes)
Den
sity
P(x > 5.5) = .5(.5) = .25
![Page 21: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/21.jpg)
Application Problem Continued . . .What is the probability that it takes more than 5.5 minutes to process the application form?
0.5
4 5 6Time (in minutes)
Den
sity
P(x > 5.5) = .5(.5) = .25When the density is constant over an interval (resulting in a horizontal density curve), the probability distribution is called a uniform
distribution.
![Page 22: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/22.jpg)
Other Density CurvesSome density curves resemble the one below. Integral calculus is used to find the area under the these curves.Don’t worry – we will use tables (with the values already calculated). We can also use calculators or statistical software to find the area.
![Page 23: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/23.jpg)
The probability that a continuous random variable x lies between a lower limit a and an upper limit b is
P(a < x < b) = (cumulative area to the left of b) –(cumulative area to the left of a)
P(a < x < b) = P(x < b) – P(x < a)
![Page 24: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/24.jpg)
Let X be the amount of time (in minutes) that aparticular San Francisco commuter must wait for aBART train. Suppose that the density curve is as pictured (a uniform distribution):
a. What is the probability that x is less than 10 minutes? more than 15 minutes?
![Page 25: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/25.jpg)
a. What is the probability that x is less than 10 minutes? more than 15 minutes?
![Page 26: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/26.jpg)
b. What is the probability that X is between 7 and 12 minutes?
![Page 27: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/27.jpg)
c. Find the value c for which P(x < c) = .9.
![Page 28: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/28.jpg)
The graph gives the distribution of the yearly amount of rainfall in Rainy City:
In a randomly selected year,(a) What is the probability that Rainy City got more
than eight inches of rain?
P(Rainy City got more than 8 inches of rain)= P(X > 8) = 1 – P(X < 8) = 1 – [0.44 + 0.30 + 0.15 + 0.06] = 1 – 0.95 = 0.05
![Page 29: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/29.jpg)
The graph gives the distribution of the yearly amount of rainfall in Rainy City:
In a randomly selected year,(b) What is the probability that Rainy City got between two and six inches of rain?
P(Rainy City got between 2 and 6 inches of rain)= P(2 < X < 6) = 0.30 + 0.15 = 0.45
![Page 30: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/30.jpg)
The graph gives the distribution of the yearly amount of rainfall in Rainy City:
In a randomly selected year,(c) What is the probability that Rainy City got
exactly two inches of rain?P(Rainy City got exactly 2 inches of rain) = 0
![Page 31: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/31.jpg)
The graph gives the distribution of the yearly amount of rainfall in Rainy City:
In a randomly selected year,(d) What is the probability that Rainy City got at most six inches of rain?P(Rainy City got at most 6 inches of rain)= P(X ≤ 6) = 0.44 + 0.30 + 0.15 = 0.89
![Page 32: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/32.jpg)
7.3 Objectives:ü Understand the definition and properties of continuous random variablesü Be able to represent the probability distributions of continuous random variables using tables, formulae, or histograms
![Page 33: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/33.jpg)
7.3 Objectives:ü Be able to interpret formulaic or graphic representations of probability distributions of continuous random variables.ü Be able to calculate probabilities such as P(x < a), P(x ≥ a), and P(a ≤ x ≤ b)
![Page 34: Chapter 7.3 - Probability Distributions for Continuous ... · Probability Distributions for Continuous Variables • Is specified by a curve called a density curve. • The function](https://reader034.vdocument.in/reader034/viewer/2022052206/5f1c62845f70284fa10f9cdb/html5/thumbnails/34.jpg)
• HW: Read 7.3: Probability Distributions for Continuous Random Variables
For Tonight:
P414: 7.20, 7.21, 7.22, 7.23, 7.24, 7.26