1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables...
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Transcript of 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables...
![Page 1: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/1.jpg)
Lecture 6
1Outline
1. Random Variablesa. Discrete Random Variables
b. Continuous Random Variables
2. Symmetric Distributions
3. Normal Distributions
4. The Standard Normal Distribution
![Page 2: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/2.jpg)
Lecture 6
21. Random Variables
Two kinds of random variables:
a. Discrete (DRV) Outcomes have countable values Possible values can be listed E.g., # of people in this room
Possible values can be listed: might be …28 or 29 or 30…
![Page 3: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/3.jpg)
Lecture 6
31. Random Variables
Two kinds of random variables:
b. Continuous (CRV) Not countable Consists of points in an interval E.g., time till coffee break
![Page 4: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/4.jpg)
Lecture 6
41. Random Variables
The form of the probability distribution for a CRV is a smooth curve. Such a distribution may also be called a
Frequency Distribution Probability Density Function
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Lecture 6
51. Random Variables
In the graph of a CRV, the X axis is whatever you are measuring (e.g., exam scores, depression scores, # of widgets produced per hour).
The Y axis measures the frequency of scores.
![Page 6: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/6.jpg)
Lecture 6
6
X
The Y-axis measures frequency. It is usually not shown.
![Page 7: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/7.jpg)
Lecture 6
72. Symmetric Distributions
In a symmetric CRV, 50% of the area under the curve is in each half of the distribution.
P(x ≤ ) = P(x ≥ ) = .5
Note: Because points are infinitely thin, we can only measure the probability of intervals of X values – not of individual X values.
![Page 8: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/8.jpg)
Lecture 6
8
µ
50% of area
![Page 9: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/9.jpg)
Lecture 6
93. Normal Distributions
A particularly important set of CRVs have probability distributions of a particular shape: mound-shaped and symmetric. These are “normal distributions”
Many naturally-occurring variables are normally distributed.
![Page 10: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/10.jpg)
Lecture 6
10Normal Distributions
are perfectly symmetrical around their mean, .
have the standard deviation, , which measures the “spread” of a distribution – an index of variability around the mean.
![Page 11: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/11.jpg)
Lecture 6
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µ
![Page 12: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/12.jpg)
Lecture 6
12Standard Normal Distribution
The area under the curve between and some value X ≥ has been calculated for the “standard normal distribution” and is given in the Z table (Table IV).
E.g., for Z = 1.62, area = .4474
(Note that for the mean, Z = 0.)
![Page 13: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/13.jpg)
Lecture 6
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XZ = 1.62Z = 0
Area gives the probability of finding a score between the mean and X when you make an observation
.4474
![Page 14: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/14.jpg)
Lecture 6
14Using the Standard Normal Distribution
Suppose average height for Canadian women is 160 cm, with = 15 cm.
What is the probability that the next Canadian woman we meet is more than 175 cm tall?
Note that this is a question about a single case and that it specifies an interval.
![Page 15: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/15.jpg)
Lecture 6
15Using the Standard Normal Distribution
160 175
We need this areaTable gives this area
![Page 16: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/16.jpg)
Lecture 6
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Remember that area above the mean, , is half (.5) of the distribution.
µ
![Page 17: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/17.jpg)
Lecture 6
17Using the Standard Normal Distribution
160 175
Call this shaded area P. We can get P from Table IV
![Page 18: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/18.jpg)
Lecture 6
18Using the Standard Normal Distribution
Z = X - = 175-160
15
= 1.00
Now, look up Z = 1.00 in the table.
Corresponding area (= probability) is P = .3413.
![Page 19: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/19.jpg)
Lecture 6
19Using the Standard Normal Distribution
160 175
This area is .3413
So this area must be .5 – .3413 = .1587
![Page 20: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/20.jpg)
Lecture 6
20Using the Standard Normal Distribution
Z = 0 Z = 1.0
This area is .3413
So this area must be .5 – .3413 = .1587
![Page 21: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/21.jpg)
Lecture 6
21Using the Standard Normal Distribution
What is the probability that the next Canadian woman we meet is more than 175 cm tall?
Answer: .1587
![Page 22: 1 Lecture 6 Outline 1. Random Variables a. Discrete Random Variables b. Continuous Random Variables 2. Symmetric Distributions 3. Normal Distributions.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649f335503460f94c4f4e8/html5/thumbnails/22.jpg)
Lecture 6
22Review
Area under curve gives probability of finding X in a given interval. Area under the curve for Standard Normal Distribution is given in Table IV. For area under the curve for other normally-distributed variables first compute:
Z = X -
Then look up Z in Table IV.