This document discusses key concepts in joint random variables including joint and marginal probability distributions, conditional probability, independence, covariance, and correlation. It provides examples of how to calculate these measures using random variables like the number of mobile data bars and website response time. The goal is to describe how multiple random variables vary together and the relationship between them.

1. MATHEMATICS IV: MA202
LECTURE 7

2. JOINT RANDOM VARIABLES|EXAMPLE
• The response time is the speed of page
downloads and it is critical for a mobile Web
site.
• As the response time increases, customers
become more frustrated and potentially
abandon the site for a competitive one.
• Let X denote the number of bars of service on
the mobile device.
• Let Y denote the response time (to the nearest
second) for a particular user and site.

3. JOINT PROBABILITY MASS FUNCTION
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4. JOINT PROBABILITY DENSITY FUNCTION
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5. EXAMPLE |SERVER ACCESS TIME
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6. MARGINAL PROBABILITY DISTRIBUTION |
EXAMPLE
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7. MARGINAL PROBABILITY DENSITY FUNCTION
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8. MARGINAL PROBABILITY DENSITY FUNCTION|
EXAMPLE SERVER ACCESS TIME
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9. CONDITIONAL PROBABILITY
DISTRIBUTION |EXAMPLE
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10. CONDITIONAL PROBABILITY DENSITY
FUNCTION
• •

11. CONDITIONAL PROBABILITY| EXAMPLE
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12. CONDITIONAL MEAN AND VARIANCE |
EXAMPLE
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13. INDEPENDENCE
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14. INDEPENDENCE EXAMPLE
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15. COVARIANCE
• •
When two or more random variables are defined on a
probability space, it is useful to describe how they vary
together.

16. COVARIANCE
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17. COVARIANCE
Zero Covariance does not imply independence!

18. CORRELATION
There is another measure of the relationship
between two random variables that is usually
easier to interpret than covariance.
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