Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values
description
Transcript of Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values
![Page 1: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/1.jpg)
Ch4:
4.1 Probability Density Function (pdf)
4.2 CDFs and expected values
![Page 2: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/2.jpg)
Continuous pdfs, CDFs and Expectation Section 4.1-2
A Random Variable: is a function on the outcomes of an experiment; i.e. a function on outcomes in S.
A discrete random variable is one with a sample space that is finite or countably infinite. (Countably infinite => infinite yet can be matched to the integer line)
A continuous random variable is one with a continuous sample space.
![Page 3: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/3.jpg)
Still going through these steps (even if it is deep in our heart):
1) Identify the experiment of interest and understand it well (including the associated population)
2) Identify the sample space (all possible outcomes)
3) Identify an appropriate random variable that reflects what you are studying (and simple events based on this random variable)
4) Construct the probability distribution associated with the simple events based on the random variable
Continuous pdfs, CDFs and Expectation Section 4.1-2
![Page 4: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/4.jpg)
Example: Continuous data:
Grades out of 10012 77 83 87 9248 78 84 88 9455 78 84 88 9457 78 84 88 9462 79 84 89 9665 79 85 89 9767 80 85 89 9769 80 86 90 9869 80 86 90 9871 80 86 9171 80 86 9172 81 87 9173 81 87 9173 81 87 9174 82 87 9174 82 87 9176 82 87 9276 82 87 9277 82 87 92
Continuous pdfs, CDFs and Expectation Section 4.1-2
![Page 5: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/5.jpg)
Class or Bin FrequencyRelative
Frequency10-<20 1 0.0120-<30 0 030-<40 0 040-<50 1 0.0150-<60 2 0.0260-<70 5 0.0670-<80 16 0.1980-<90 39 0.46
90-<100 21 0.25 Total 85 1
Continuous pdfs, CDFs and Expectation Section 4.1-2
![Page 6: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/6.jpg)
Histogram of x
x
Density
20 40 60 80 100
0.00
0.01
0.02
0.03
0.04
Called density because the total area of the graph is 1.
Continuous pdfs, CDFs and Expectation Section 4.1-2
![Page 7: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/7.jpg)
When we are looking at the entire population then,
Continuous pdfs, CDFs and Expectation Section 4.1-2
![Page 8: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/8.jpg)
Histogram of x
x
Density
20 40 60 80 100
0.00
0.02
0.04
Histogram of x
x
Density
20 40 60 80 100
0.00
0.02
0.04
Histogram of x
x
Density
20 40 60 80 100
0.00
0.02
0.04
Histogram of x
x
Density
20 40 60 80 100
0.00
0.04
Histogram of x
x
Density
20 40 60 80 100
0.00
0.04
0.08
Histogram of x
x
Density
20 40 60 80 1000.0
0.4
0.8
Continuous pdfs, CDFs and Expectation Section 4.1-2
We can reduce the width of the class and have it approach zero.
![Page 9: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/9.jpg)
Histogram of x
x
Density
20 40 60 80 100
0.00
0.02
0.04
0.06
0.08
0.10
Continuous pdfs, CDFs and Expectation Section 4.1-2
The Height can then be found using the function that approximates the structure of the sample space. And,
Very small close to zero
Also very small close to zero
![Page 10: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/10.jpg)
Continuous pdfs, CDFs and Expectation Section 4.1-2
So the chance of observing the value x is pretty much zero!
This is the major difference from the discrete distribution structure, the rest follows:
![Page 11: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/11.jpg)
Probability distributions for discrete rvs Section 3.2
For continuous random variables, we call f(x) the probability density function (pdf).
From the axioms of probability, we can show that:
1.
2.
1.
2.Opposed to
![Page 12: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/12.jpg)
Probability distributions for discrete rvs Section 3.2
Based on the pdf we construct the CDF, F(x),
Opposed to
![Page 13: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/13.jpg)
We can also find the pdf using the CDF if we note that:
Probability distributions for discrete rvs Section 3.2
So, for any two numbers a, b where a < b,
![Page 14: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/14.jpg)
Expected values Section 3.3
The expected value E(X) of a continuous random variable is,
Compared to
![Page 15: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/15.jpg)
Expected values Section 3.3
The variance, E[(X - E(X))2] of a contunuous random variable is,
It measures the amount of variation or uncertainty in the to be observed (or the observed) value of a random variable.
![Page 16: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/16.jpg)
Expected values Section 3.3
The standard deviation,
Also measures the amount of variation or uncertainty in the to be observed (or the observed) value of a random variable. It has the same units as the random variable.
![Page 17: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/17.jpg)
Expected values Section 3.3
Some properties of expectation:Let h(X) be a function, a and b be constants then,
![Page 18: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/18.jpg)
Example: The current measured in a thin wire has an equal chance of being in an the interval [1, 10] milliampere. What is the probability that at a certain time the measured current is more than 7 mA’s? What is the mean current over time? The variance?
Continuous pdfs, CDFs and Expectation Section 4.1-2
![Page 19: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/19.jpg)
1) Identify the experiment of interest and understand it well
2) Identify the sample space
3) Identify an appropriate random variable that reflects what you are studying
Continuous pdfs, CDFs and Expectation Section 4.1-2
![Page 20: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/20.jpg)
4) Construct the probability distribution associated with the simple events based on the random variable
Continuous pdfs, CDFs and Expectation Section 4.1-2
0 2 4 6 8 10
0.00
0.02
0.04
0.06
0.08
0.10
x
density
![Page 21: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/21.jpg)
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
x
F(x)
4) Construct the probability distribution associated with the simple events based on the random variable
Continuous pdfs, CDFs and Expectation Section 4.1-2
![Page 22: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/22.jpg)
The Uniform distribution
Continuous pdfs, CDFs and Expectation Section 4.1-2
![Page 23: Ch4: 4.1Probability Density Function ( pdf ) 4.2CDFs and expected values](https://reader035.fdocuments.us/reader035/viewer/2022062520/56815f2f550346895dcdfd83/html5/thumbnails/23.jpg)
Example: Observing the time until a red car passes through the main and sixth intersection.
S =
Random variable X = time till we observe a red car go through main and 6th. X(0) = 0, X(0.0001) = 0.0001
A one-to-one transformation of S.
Probability distribution:
Continuous random variable! Sample space is continuous.
Continuous pdfs, CDFs and Expectation Section 4.1-2