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Discrete Random Variable

Discrete Random Variable. Outline. Expected Value (Section 2.5 ) Functions of a Random Variable (Section 2.6) Expected Value of a Derived Random Variable (Section 2.7) Variance and Standard of Deviation (Section 2.8) Conditional Probability Mass Function (Section 2.9). Expected Value.

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Discrete Random Variable

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  1. Discrete Random Variable

  2. Outline • Expected Value (Section 2.5) • Functions of a Random Variable (Section 2.6) • Expected Value of a Derived Random Variable (Section 2.7) • Variance and Standard of Deviation (Section 2.8) • Conditional Probability Mass Function (Section 2.9)

  3. Expected Value

  4. Expected Value

  5. Expected Value of a Geometric Random Variable

  6. Functions for generating random samples • bernoullirv(p,m) • binomialrv(p,m) • geometricrf(p,m) • pascalrv(k,p,m) • poissonrv(alpha,m)

  7. mean() in matlab • mean() calculates the average of a vector. • E[X] is not the same as mean(X) unless the length of the vector is sufficiently large.

  8. Calculate the Expected Value of a Geometric RV in Matlab • X is a geometrical random variable. • Analytical calculation: E[X]=5 for p=0.2

  9. Functions of a Random Variable

  10. Derived Random Variable • We observe sample values of a random variable and these sample values to compute other quantities • Example: • We measure the power level of received signal in a cellular phone. An observation is x, the power level in units of milliwatts. • We convert the measurements to decibles by calculating y=10log10(x)dBm • X is a random variable • Y is a derived random variable because it is derived from X

  11. 1 page: 10 cents 2 pages: 19 cents 3 pages: 27 cents 4 pages: 34 cents 5 pages: 40 cents Determine Y with a polynomial: Y=aX+bX2 Determine a and b using X=1,3. Solve the equations in Mathematica or by hand.

  12. Simulate PC(c) in matlab

  13. Expected Value of a Derived Random Variable

  14. Expected Value of a Derived Variable

  15. Example • Prove the following theorem:

  16. Estimating Expected Monthly Phone Bill

  17. Variance and Standard Deviation

  18. Key Points • An average is a typical value of a random variable. • The next question: • “How typical is the average?” • “What are the chances of observing an event far from the average?” • The variance of a random variable X describes the difference between X and its expected value.

  19. Definitions

  20. Properties of Variance/Standard of Deviation

  21. Conditional Probability Mass Function

  22. Definition/Theorems

  23. If time permits…..

  24. Matlab

  25. Generating Random Variable

  26. Derived Random Variable

  27. Conditioning

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