CHAPTER 6: RANDOM VARIABLES AND EXPECTATION Mr. Mark Anthony Garcia, M.S. Mathematics Department De La Salle University CHAPTER 6: RANDOM VARIABLES AND EXPECTATION

Situation: Random Variable Consider the experiment of tossing three coins. Then the sample space S will contain the elements HHH, HHT, HTH, THH, TTH, THT, HTT and TTT. Let X be a variable representing the number of heads that occur in each outcome.

Situation: Random Variable The following table shows the assignment of values to each outcome. Outcome X HHH 3 TTT HHT 2 TTH 1 HTH THT THH HTT

Situation: Random Variable The only possible values for X are 0, 1, 2 and 3. The variable X is viewed as a random variable.

Random Variable A random variable is a function that associates a real number with each element in the sample space.

Probability Distribution A probability distribution shows the probability of each value x of the random variable X.

Example 1: Probability Distribution Consider the experiment of tossing three coins and let X be the random variable representing the number of heads that occur in each outcome. Outcome X HHH 3 TTT HHT 2 TTH 1 HTH THT THH HTT

Example 1: Probability Distribution The probability distribution of X is given in the following table. x 1 2 3 P(X=x) 1/8 3/8

Example 1: Probability Distribution The probability that there is no head in the outcome (TTT) is P(X=0) = 1/8. The probability that there is exactly one head in the outcome (HTT, THT, TTH) is P(X=1) = 3/8.

Example 1: Probability Distribution The probability that there is exactly two heads in the outcome (HHT, HTH, THH) is P(X=2) = 3/8. The probability that the outcome contains all heads (HHH) is P(X=3) = 1/8.

Properties of Probability Distribution Each probability P(X=x) is greater than 0. The sum of all probabilities is equal to 1.

Example 2: Probability Distribution Consider the experiment of rolling a pair of dice. Let X be the random representing the sum of the top faces of the pair of dice.

Example 2: Probability Distribution The following table shows the probability distribution for X. x P(X=x) 2 1/36 6 5/36 10 3/36 3 2/36 7 6/36 11 4 8 12 5 4/36 9

Situation Suppose that two coins are tossed 16 times. The following table gives the result: HT HH TT TH

Situation Let X be the random variable representing the number heads per toss. HT 1 HH 2 TT TH

Situation What is the average number of heads per toss of the two coins? 1+0+2+1+2+1+2+0+0+1+2+1+1+0+1+2 16 17 16 =1.0625

Situation We may view this average as the mathematical expectation. This means that for every toss of two coins, we expect that there would be 1 head in the outcome. However, this result is from an experiment.

Situation Moreover, we can write the solution in the following form. 0 4 16 +1 7 16 +2 5 16 = 17 16 =1.0625 x 1 2 P(X=x) 4/16 7/16 5/16

Mathematical Expectation Let X be a random variable with probability distribution 𝑃(𝑋=𝑥). The mean or expected value of X is given by 𝐸 𝑋 = [𝑥∙𝑓 𝑥 ] .

Example 3: Expectation Suppose that in a game of tossing two coins, you win PhP100.00 when two heads come out and you lose Php50.00 otherwise. What is your expected gain?

Example 3: Expectation Let X be the random variable representing the player’s gain per toss. Player wins Php100 when HH comes out. Otherwise, player loses Php50 when HT, TH and TT come out

Example 3: Expectation x Php100 -Php50 P(X=x) 1/4 3/4 The probability distribution table is shown below. Using mathematical expectation, we have 𝐸 𝑋 = 100 1 4 + −50 3 4 =−𝑃ℎ𝑝12.50. x Php100 -Php50 P(X=x) 1/4 3/4

Example 4: Expectation x Php50 -Php30 P(X=x) 13/52 39/52 Suppose that in a game, you win PhP50.00 when you draw a heart card and you lose PhP30.00 otherwise. What is your expected gain? x Php50 -Php30 P(X=x) 13/52 39/52

Example 4: Expectation The expected gain (loss) of the player is given by 𝐸 𝑋 = 50 13 52 +(−30) 39 52 . So, 𝐸 𝑋 =−𝑃ℎ𝑝10.

Example 5: Expectation By investing in a particular stock, a person can make a profit in one year of $4000 with probability 0.3 or take a loss of $1000 with probability 0.7. What is this person’s expected gain? 𝐸 𝑋 = 4000 0.3 +(−1000)(0.7) 𝐸 𝑋 =$500

Exercises In a gambling game, a man is paid $5 if he gets all heads or all tails when three coins are tossed, and he will pay out $3 if either one or two heads show. What is his expected gain?

Exercises In a gambling game, a woman is paid $3 if she draws a jack or a queen and $5 if she draws a king or an ace from an ordinary deck of 52 playing cards. If she draws any other card, she loses. How much should she pay to play if the game is fair?

Exercises A game in a TV show has two parts. In the first part, the contestants will play an elimination game and the winner will gain 50,000 pesos. The winner will have a chance to spin a spinner with 8 possible winnings: return 50,000 pesos, divide 50,000 by 2, win 10,000, win 25,000, win 50,000, win 100,000, win 250,000 and win 500,000.

Exercises Let X represents the random variable representing the total winnings of the contestant. Construct a probability distribution table X. What is the expected amount of winnings of the contestant?