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MATH 256 Probability and Random Processes

OKAN UNIVERSIT Y FACULTY OF ENGINEERING AND ARCHITECTURE. MATH 256 Probability and Random Processes. 04 Random Variables. Yrd . Doç . Dr. Didem Kivanc Tureli didemk@ieee.org didem.kivanc@okan.edu.tr. Fall 2011. Probability Mass Function. Is defined for a discrete variable X.

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MATH 256 Probability and Random Processes

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  1. OKAN UNIVERSITYFACULTY OF ENGINEERING AND ARCHITECTURE MATH 256 Probability and Random Processes 04 Random Variables Yrd. Doç. Dr. Didem Kivanc Tureli didemk@ieee.org didem.kivanc@okan.edu.tr Fall 2011 Lecture 3

  2. Probability Mass Function • Is defined for a discrete variable X. • Suppose that • Then since x must be one of the values xi, Lecture 3

  3. Example of probability mass function Lecture 3

  4. Expectation of a random variable • If X is a discrete random variable having a probability mass function p(x) then the expectation or the expected value of X denoted by E[X] is defined by • In other words, • Take every possible value for X • Multiply it by the probability of getting that value • Add the result. Lecture 3

  5. Examples of expectation • For example, suppose you have a fair coin. You flip the coin, and define a random variable X such that • If the coin lands heads, X = 1 • If the coin lands tails, X = 2 • Then the probability mass function of X is given by Or we can write Lecture 3

  6. Expectation of a function of a random variable • To find E[g(x)], that is, the expectation of g(X) • Two step process: • find the pmf of g(x) • find E[g(x)] Lecture 3

  7. Compute Solution Let Y = X 2. Then the probability mass function of Y is given by Let X denote a random variable that takes on any of the values –1, 0, and 1 with respective probabilities Lecture 3

  8. Proposition 4.1 If X is a discrete random variable that takes on one of the values xi, i ≥ 1 with respective probabilities p(xi), then any real valued function g. Check if this holds for the previous example: Lecture 3

  9. Proof of Proposition 4.1 Lecture 3

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  14. Variance Lecture 3

  15. Variance Lecture 3

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  17. Some more comments about variance Lecture 3

  18. Bernoulli Random Variables Lecture 3

  19. Binomial Random Variables Lecture 3

  20. Poisson Random Variable Some examples of random variables that generally obey the Poisson probability law [that is, they obey Equation (7.1)] are as follows: 1. The number of misprints on a page (or a group of pages) of a book 2. The number of people in a community who survive to age 100 3. The number of wrong telephone numbers that are dialed in a day 4. The number of packages of dog biscuits sold in a particular store each day 5. The number of customers entering a post office on a given day 6. The number of vacancies occurring during a year in the federal judicial system 7. The number of α-particles discharged in a fixed period of time from some radioactive material Lecture 3

  21. Expectation of the sum of r.v.s • Start with this: • It’s really quite easy to show that: • And from this we show that Lecture 3

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