C HAPTER 3 Discrete Random Variables

1. CHAPTER 3 Discrete Random Variables Prof. Sang-Jo Yoo sjyoo@inha.ac.kr http://multinet.inha.ac.kr

2. What we are going to study? • Random variable • A function that assigns a numerical value to the outcome of the experiment. • Contents • Concept of a random variable • Methods for calculating probabilities of events involving a random variable. • Probability mass function • Expected value of random variable. • Conditional probability mass function given partial information about the random variable.

3. Random Variable (1/2) • Usually interested not in the outcome itself, but numerical attribute of the outcome. • Number of heads in n tosses of a coin. • The weight of a randomly selected student. • A measurement assigns a numerical value to the outcome of the random experiment. • Since the outcomes are random, the results of the measurements will also be random. • Random variable X • A function that assigns a real number, X(ζ), to each outcome ζ in the sample space of a random experiment. • 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. Note that SX ⊂ R, R is set of all real numbers.

4. Random Variable (2/2) • Example 3.1 • A coin is tossed 3 times. Let X be the number of heads in 3 tosses • Outcomes ζ? Random variables X(ζ) ? The probability of the event {X=k}? • Sample space S ={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.Let X be the number of heads; then SX= {0, 1, 2, 3}.

5. Equivalent Events (1) • Sx= the set of values that can be taken on by X • B = the subset of Sx • A = {ζ: X(ζ) in B}: the set of outcomes ζ in S that lead to values X(ζ) in B • Since event B in SX occurs whenever event A in S occurs, and vice versa. Hence P[B] = P[A] = P[{ ζ: X( ζ ) in B}]. A and B are called equivalent eventswith respect to X.

6. Equivalent events(2) • Example • Consider the random experiment of tossing 3 coinsS ={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} • X = no of heads in the 3 coins, SX = {0, 1, 2, 3} • A1 = {HTT, TTT} • A2 = {HHT, HTH, THH, TTT} • A3 = {HTT, THT, TTH, TTT} X(A1)={0, 1} = set of all values taken by X(ζ ), ζ∈ A1 X(A2) = {0, 2} X-1({0, 1}) = set of all outcomes of elements in {0, 1}= {HTT, THT, TTH, TTT} = A3. X-1({2, 3}) = {HHT, HTH, THH, HHH} = set of all outcomes of elements in B = {2, 3}.

7. Discrete Random Variables • Discrete random variable X • A random variable that assumes values from a countable set, that is • If its range is finite, then a discrete random variable X is said to be finite. • We are interested in finding the probabilities of events involving a discrete random variable X. • Probability mass function (pmf)of a discrete random variable X • Note 3-1

8. Probability Mass Function PMF pX(x) satisfies three properties: Example 3.5 (Coin Tosses and Binomial Random Variable)

9. Bernoulli Random Variable Example 3.8 (Bernoulli Random Variable) • Let A be an event related to the outcomes of some random experiment. Indicator function for A is defined by • Identify IA =1 with a “success”. • IA is a discrete random variable. In this case, SI=range of IA={0,1}. • Let P [A ]=p, then pmf is PI (0)=1-p , PI (1)= p . • Geometric Random

10. Geometric Random Variable Count M independent Bernoulli trials until the first occurrence of a success. M is the geometric random variable. Example 3.9 (Message Transmissions) Probability mass function Cumulative distribution function

11. Binomial Random Variable A random experiment is repeated n times Let X be the number of times an event A occurs in n trials Example 3.10 (Transmission Errors) Probability mass function

12. Expected Value of Random Variables • 15 repetitions of a random experiment • X varies about 5, Y varies about 0 • X is more spread out than Y • Need parameters that quantify these properties

13. Expected Value of a RV X • The expected value of a discrete RV X • The expected value is defined if the above integral or sum converges absolutely, • Example 3.11 (Mean of Bernoulli Random Variable) • Example 3.12 (Three Coin Tossses and Binomial Random Variable)

14. Expected Value of the Geometric Random Variable • Can we find a closed form for the above summed series? • Recall • Hence, • Example • The chance of getting “6” in the throw of a dice is 1/6. The expected number of trials required to get the first “6” is . Does the answer sound reasonable?

15. Expected Value of Functions of a RV Example 3.17 Example 3.19 • Let X be a discrete RV, and let • Z will assume a countable set of values of the form • Group the term xk that are mapped to each value zj • Let Z be the function

16. Variance of a Random Variable (1/2) • The expected value provides us with very limited information. • The deviation of about its mean. • The variance of X • The standard deviation of X

17. Variance of a Random Variable (2/2) • Properties of variance • The nthmoment of the random variable X

18. Conditional Probability Mass Function • In man situations we have partial information about a random variable X or about the outcome of its underlying random experiment. • We are interested in how this information changes the probability of events involving the random variable. • Conditional probability mass function • Conditional expected value • Note 3-2

19. Important Discrete Random Variables VAR 1/4 p 1/2 1. Bernoulli Random Variable • Remark • The Bernoulli RV is the value of the indicator function IA for some event A; X=1 if A occurs and 0 otherwise. • The variance is quadratic in p, with value zero at p=0 and p=1 and maximum at p=1/2.

20. Important Discrete Random Variables 2. Binomial Random Variable • Remark: X is the number of successes in n Bernoulli trials and hence the sum of n independent, identically distributed Bernoulli RV.

21. Variance of Binomial RV

22. Important Discrete Random Variables Example 3.30 3. Geometric Random Variable • Remark: X is the number of trials until the first success in a sequence of independent Bernoulli trials. Using Setting and multiplying both side by p

23. Variance of Geometric RV Using Setting x=q and multiplying both sides by pq, we obtain Since • Therefore,

24. Memoryless property • The discrete geometric random variable observes the memoryless property: • If a success has not occurred in the earlier j trials, then the probability of having to perform at least k more trials to get a success is the same as the probability of initially having to perform at least k trials to get a success. • Proof • First, observe that • We then have

25. Important Discrete Random Variables 4. Poisson Random Variable • Remark: Counting the number of occurrences of an event in a certain time period or a certain region in space, e.g. counts of emissions from radioactive substances. • The pmf for a Poisson random variable N is • where is the average number of event occurrences in a specified time interval or region in space. • Note