This document discusses methods for deriving probability mass functions from cumulative distribution functions. It explains that the probability mass function at each point with non-zero probability is equal to the jump in the cumulative distribution function at that point. It also discusses differentiating the cumulative distribution function to determine the probability density function of a continuous random variable. Additionally, it covers calculating the mean and variance of random variables.

1. MA202: MATHEMATICS IV
Lecture 4

2. Cumulative Distribution Functions
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3. Going back from cumulative distribution
(cdf) to probability mass function (pmf)!
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What are the points with non-zero probability?
The probability mass function at each point is
the jump in the cumulative distribution function
at that point.

4. Going back from cumulative distribution
(cdf) to probability mass function (pmf)!
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5. Continuous Random Variable | Probability
distributions |Probability Density Functions
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6. Continuous Random Variable |
Probability Density Functions | Example
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7. Continuous Random Variable |
Probability Density Functions | Example
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8. Continuous Random Variable |
Probability Density Functions
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9. Cumulative Distribution Function|
Continuous Random Variable
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10. CDF to PDF
► The probability density function of a continuous random variable can be
determined from the cumulative distribution function by differentiating.
► Note that the cdf of a continuous random variable is always continuous unlike
that of a discrete random variable.

11. CDF to PDF| Example
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12. Mean and Variance of a Random Variable
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13. Mean and Variance of a Random Variable
|Example
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10 11 12 13 14 15
0.08 0.15 0.30 0.20 0.20 0.07

14. Mean and Variance of a Random Variable
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15. Mean and Variance of a Random
Variable| Example
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16. Mean and Variance of a Random
Variable| Example
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