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Chapter 16: Random Variables Random Variables A numerical variable whose value is based on the outcome of a random event is called a random variable • if we can list the possible values of the variable it is called discrete, otherwise it is called continuous • x = value rolled on a die is discrete • x = how long it takes to drive to work on any given day is continuous • The collection of all possible values and the probabilities of each occurring is called the probability model of the variable • x = the value rolled on a standard 6-sided die x P(x) 1 1 2 1 3 1 4 1 5 1 6 1 /6 ≈ 0.1667 /6 ≈ 0.1667 /6 ≈ 0.1667 /6 ≈ 0.1667 /6 ≈ 0.1667 /6 ≈ 0.1667 Example 1: Carnival Game #1 You roll a 6-sided die. If it comes up 5 you win $100. If not you roll again. If it comes up 5, you win $50. If not, you pay $20. Let x = amount you “win” when you play this game once. Give the probability model of x. Def. The mean or expected value of a random variable x represents the mean of the numerical data set that could be created by observing the value of a random variable many, many, many times (an infinite number of times). This is denoted m x , m ( x ), or E ( x ) . The standard deviation of a random variable represents the standard deviation of the numerical data set described above. This is denoted s x or s ( x ) . • mx = å x × P ( x) • sx = å( x - m ) 2 x × P ( x) Example 2: Carnival Game #1 Revisited Find the m x and s x for the random variable described in Example 1. This can also be done via the TI calculator. Put the values of x in L1 and the corresponding probabilities in L2, and the run the 1-Var Stats command with L1, L2 next to it: Press Press then enter the values in L1 and L2 Example 3: Carnival Game #2 A carnival game offers a $100 cash prize for anyone who can break a balloon by throwing a dart at it from a certain distance. It costs $5 to play, and you’re willing to spend up to $20 trying to win, but if you win before spending $20, you will stop. Let x = the amount you profit from this game, and suppose there is a 10% chance of breaking the balloon on any one throw. Give the distribution of x. Now work on Random Variables Practice handout