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Expectation Random Variables Graphs and Histograms Expected Value Random Variables A random variable is a rule that assigns a numerical value to each outcome of an experiment. We will classify random variables as either Finite discrete – if it can take on only finitely many possible values. Infinite discrete – infinitely many values that can be arranged in a sequence. Continuous – if its possible values form an entire interval of numbers Example One Suppose that we toss a fair coin three times. Let the (finite discrete) random variable X denote the number of heads that occur in three tosses. Then Sample Pt. Value of X Sample Pt. Value of X HHH HHT HTH 3 2 2 HTT THT TTH 1 1 1 THH 2 TTT 0 Example Two Suppose that we toss a coin repeatedly until a head occurs. Let the (infinite discrete) random variable Y denote the number of trials. Sample Pt. Value of Y Sample Pt. Value of Y H TH TTH 1 2 3 TTTTH TTTTTH TTTTTTH 5 6 7 TTTH 4 Example Three A biologist records the length of life (in hours) of a fruit fly. Let the (continuous) random variable Z denote the number of hours recorded. If we assume for simplicity, that time can be recorded with perfect accuracy, then the value of Z can take on any nonnegative real number. Graphs and Histograms Given a random variable X, we will be interested in the probability that X takes on a particular real value x, symbolically we write pX(x) = P(X = x) pX(x) is referred to as the probability function of the random variable X. Geometric Representation Consider Example Two where a coin is tossed three times. From the given table we see that Histogram p(0)= P(X= 0)= 1/8 p(1)= P(X= 1)= 3/8 p(2)= P(X= 2)= 3/8 p(3)= P(X= 3)= 1/8 P(x)-values 4/8 3/8 2/8 1/8 0 0 1 2 x-values 3 Line and Bar Graphs Bar Graph Line Graph 4/8 p(x)-values p(x)-values 4/8 3/8 2/8 ` 1/8 3/8 2/8 ` 1/8 0 0 0 1 2 x-values 4 0 1 2 x-values 4 Expectation x1 x2 xn Arithmetic Mean x n Consider 10 hypothetical test scores: 65, 90, 70, 65, 70, 90, 80, 65, 90, 90 Calculate the mean as follows: 3 65 2 70 1 80 4 90 3 2 1 4 x 65 70 80 90 77.5 10 10 10 10 10 Expectation We may express the arithmetic mean as: fk f1 f2 x x1 x2 xk n n n As the number of repetition increases fi xi pi n