Random Variables A Random Variable assigns a numerical value to all possible outcomes of a random experiment We do not consider the actual events but we associate numbers with the events that arise from the experiment

Random Variables, con’t Consider the experiment of tossing a fair coin three times What is the sample space associated with this experiment:  TTT,TTH,THT,HTT,HHT,HTH,THH,HHH  In this case, we may say that our random variable, X, records the number of heads in three tosses What are the possible values for X? In this case, the random variable is used to describe certain events.

Random Variables, con’t In general, given an experiment and a random variable, X, and a number, x, the expression X=x stands for the event that the random variable takes on the value of x X=0 is the event that no heads occur,  TTT  X=1 is the event that exactly one head occurs  TTH,THT,HTT  X=2 is the event that exactly two heads occur  HHT,HTH,THH  X=3 is the event that exactly three head occurs  HHH 

Random Variables, con’t Since the number, X, that is associated with outcomes takes on different values by chance, it is called a random variable.

Random Variables, con’t  TTT,TTH,THT,HTT,HHT,HTH,THH,HHH  We let P(X=x) be the probability that the event X=x occurs What is P(X=0)? What is P(X=1)? What is P(X=2)? What is P(X=3)?

Random Variables, con’t Let’s make a table with all possible values of x and the corresponding probabilities Notice that the probabilities sum to 1 xP(X=x)P(X=x) 01/8 13/8 2 31/8 Total8/8=1

Random Variables, con’t  TTT,TTH,THT,HTT,HHT,HTH,THH,HHH  We may also have probabilities such as: P(X  x), P(X  x), P(a  X  b), P(X  x) What is P(0  X  1)? What is P(0  X<2)? What is P(X  3)? What is P(X  5)?

Expected Value of a Random Variable Let’s play a game: We will roll a fair die. If we roll a 1 or 2 you lose $5, if we roll a 3 or 4 you don’t owe me anything, if we roll a 5 or 6 you win $20. Who wants to play one time? Who wants to play three times? Who wants to play ten times?

Expected Value of a Random Variable The probability experiment is tossing a die Sample space  1, 2, 3, 4, 5, 6  Let’s think about the game: Let X be the amount won on one play of the game, what values can X take on? xP(X=x)

Expected Value of a Random Variable, con’t E(X) is the Expected value of a random variable X E(X) is the average value that the random variable, X, would take on after infinitely many trials Suppose that X can assume only the distinct values x 1, x 2,..., x n. The expected value of X, denoted by E(X), is the sum of these values, weighted by their respective probabilities. That is,

Expected Value of a Random Variable, con’t Let’s look at the expected value or the average amount that you would win over the long run xP(X=x)P(X=x) xP(x=x)xP(x=x)