1. Discrete Random Variable Random Process

2. The Notion of A Random Variable We expect some measurement or numerical attribute of the outcome of a random experiment A measurement assigns a numerical value to the outcome of the random experiment The outcomes are random, so are the results of the measurements A random variable is a function that assigns a real number, to each outcome in the sample space of a random experiment

3. The Notion of A Random Variable A function is a rule for assigning a numerical value to each element of a set The sample space S is the domain of the random variable, and the set SX of all values taken on by X is the range of the random variable

4. Example 1 A coin is tossed three times and the sequence of head and tails is noted. The sample space for this experiment is S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Let X be the number of heads in the three tosses We can see that X assigns each outcome in S a number from the set SX = {0, 1, 2, 3}

5. Example 2 Continued from example 1, a player receives $1 if X= 2 and $8 if X = 3, but nothing otherwise Let Y is the reward to the player, thus Y is a function of the random variable X We note that SY = {0, 1, 8}

6. Example 3 Continued from example 2, we will find the probability of the event {X = 2} We also want to find the probability that a player wins $8

7. Discrete Random Variables and Probability Mass Function A discrete random variable X is defined as a random variable that assumes values from a countable set, A finite discrete random variable is defined when the range is finite, The probability mass function (pmf) of a discrete random variable X is defines as Pmf is a function of x over the real line, nonzero only at values x1, x2, x3,...

8. Discrete Random Variables and Probability Mass Function The pmf satisfies the following properties:

9. Example 4 Let X be the number of heads in three independent tosses of a coin The pmf of X is Note that Remember Newton’s triangle

10. Example 5 From example 4, Y is a random variable for reward If a player receives $1 if the number of heads in three coin tosses is 2 and $8 if the number is 3, but nothing otherwise (see example 2) The pmf is Note that

11. Example 6: Message Transmission Let X be the number of times of a message needs to be transmitted until it arrives correctly at its destination The event occurs if the underlying experiment finds k – 1 consecutive erroneous transmissions followed by an error- free one It is called geometric random variable The probability that X is an even number is

12. Example 7: Transmission Errors A binary communications channel introduces a bit error in a transmission with probability p Let X be the number of errors in n independent transmissions The pmf of X We call it the binomial random variable The probability of one or fewer errors

13. Expected Value and Moments of Discrete Random Variable The expected value or mean of a discrete random variable is The expected value is defined if the above sum converges absolutely

14. Example 8 Let X be the number of heads in three tosses of a fair coin The expected value is Note that n = 3 and p = ½  binomial random variable Thus, the expected value is

15. Example 9 A player at a fair pays $1.50 to toss a coin three times. The player receives $1 if the number of heads is 2, $8 if the number is 3, but nothing otherwise (see example 2) The expected value is The expected gain is It means that players, on average, lose 12.5 cents per game  nice profit for the house

16. Variance of A Random Variable The variance of the random variable X is defined as OR The standard deviation of the random variable X is Second moment of X

17. Variance of A Random Variable Some useful properties o If Y = X + c, then o If Z = cX, then o If X = c (a constant with unity probability)

18. Example 10 Let X be the number of heads in three tosses of a fair coin, the variance is For special case of binomia random variable, the variance is equal to npq

19. Conditional Probability Mass Function Let X be a discrete random variable with pmf and let C be an event that has nonzero probability The conditional probability mass function of X is defined as Applying the definition of conditional probability

20. Conditional Expected Value Let X be a discrete random variable, and suppose that we know that event B has occured The conditional expected value of X given B is The conditional variance of X given B is

21. Important Discrete Random Variables Please read by yourself the types of discrete random variables o Bernoulli Random Variable o Binomial Random Variable o Geometric Random Variable o Negative Binomial Random Variable o Poisson Random Variable o Uniform Random Variable o Zipf Random Variable

22. Homeworks The problems are taken from Garcia’s book, due: next week For International Class Do problems 3.7, 3.17, and 3.30 For Regular Class Make group of 2, so there are 11 groups If n the number of group, then each group must do problems 3.n, 3.(11+n), 3.(22+n) Example: group 1 must do problems 3.1, 3.12, and 3.23 Group 11 must do problems 3.11, 3.22, and 3.33