continuous random variable (1)
DESCRIPTION
Continuous Random Variable (1). Section 3.1-3.3. Continuous Random Variable. What is the probability that X is equal to x ?. CDF for a Discrete Random Variables. Question: Is there a CDF for a continuous random variable if a PMF can not be identified for a continuous random variable?. - PowerPoint PPT PresentationTRANSCRIPT
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Continuous Random Variable (1)
Section 3.1-3.3
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Continuous Random Variable
What is the probability that X is equal to x?
![Page 3: Continuous Random Variable (1)](https://reader035.vdocuments.net/reader035/viewer/2022062314/56812a5b550346895d8dc2eb/html5/thumbnails/3.jpg)
CDF for a Discrete Random Variables
Question: Is there a CDF for a continuous random variable if a PMF can not beidentified for a continuous random variable?
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CDF for a Wheel-Spinning Experiment
P[X=≤ x]=x if 0 ≤X ≤1
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CDF for Continuous Random Variable
• Even though it is not possible to define a PDF for a continuous random variable, it is possible to define a CDF for a random variable
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PMF to CDF for a Discrete Random Variable
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Theorem 2.2
Theorem 2.3
What contributes to the jump in the CDF?
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Discrete RV
Continuous RV
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Compare CDF of a Continuous RV to that of a Discrete RV
Discrete RV:1. Zero slope2. Jumps in CDF
Continuous RV:A continuous function
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Slope of CDF function
The slope at any point x indicates the probability that X is near x.
(Just as the jump in the CDF of a discrete RV suggests non-zero probability at X=x, so does a slope in CDF of a continuous random variable?)
![Page 11: Continuous Random Variable (1)](https://reader035.vdocuments.net/reader035/viewer/2022062314/56812a5b550346895d8dc2eb/html5/thumbnails/11.jpg)
Probability Density Function (PDF)
It is not possible to define a PMF function for a continuous variable because P[X=x]=0.We can, however, define a probability density function.
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Properties of fX(x)
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PDF of X
![Page 14: Continuous Random Variable (1)](https://reader035.vdocuments.net/reader035/viewer/2022062314/56812a5b550346895d8dc2eb/html5/thumbnails/14.jpg)
Expected Value
Discrete Random Variable