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Math 145 October 3, 2006 Random Variable – A random variable is a variable whose value is a numerical outcome of a random phenomenon. – A random variable is a function or a rule that assigns a numerical value to each possible outcome of a statistical experiment. Examples Tossing a coin 3 times: Sample Space = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Random variables : X1 = The number of heads. = {3, 2, 2, 2, 1, 1, 1, 0} X2 = The number of tails. = {0, 1, 1, 1, 2, 2, 2, 3} Rolling a Pair of Dice Sample Space: (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) Rolling a Pair of Dice Random variable: X3 = Total no. of dots 2 3 4 5 6 7 3 4 5 6 7 8 4 5 6 7 8 9 5 6 7 8 9 10 6 7 8 9 10 11 7 8 9 10 11 12 Rolling a Pair of Dice X4 = (positive) difference in the no. of dots 0 1 2 3 4 5 1 0 1 2 3 4 2 1 0 1 2 3 3 2 1 0 1 2 4 3 2 1 0 1 5 4 3 2 1 0 Rolling a Pair of Dice X5 = Higher of the two. 1 2 3 4 5 6 2 2 3 4 5 6 3 3 3 4 5 6 4 4 4 4 5 6 5 5 5 5 5 6 6 6 6 6 6 6 More Examples Survey: Random variables : X6 = Age in years. X7 = {1=male, 0=female}. X8 = Height. Medical Studies: Random variables : X9 = Blood Pressure. X10 = {1=smoker, 0=non-smoker}. Probability Distribution Tossing a coin 3 times: Sample Space = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Random variable : X1 = The number of heads. = {3, 2, 2, 2, 1, 1, 1, 0} x Prob. 0 1/8 1 3/8 2 3/8 3 1/8 Probability Histogram Tossing a coin 3 times: Random variable : X1 = The number of heads. X 0 1 2 3 Prob. 1/8 3/8 3/8 1/8 1/2 3/8 1/4 Prob 1/8 0 0 1 2 3 Rolling a Pair of Dice Sample Space: (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) Rolling a Pair of Dice Random variable: X3 = Total no. of dots x P 2 2 3 4 5 3 4 5 6 4 5 6 7 5 6 7 8 6 7 8 9 7 8 9 10 6 7 7 8 8 9 9 10 10 11 11 12 3 4 5 6 7 8 9 10 11 12 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 Rolling a Pair of Dice Random variable: X3 = Total no. of dots x P 2 3 4 5 6 7 8 9 10 11 12 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 1/4 1/6 5/36 1/9 1/8 Prob 5/36 1/9 1/12 1/12 1/18 1/18 1/36 1/36 0 2 3 1. Pr(X3<5)= 4 5 6 7 8 9 10 11 2. Pr(3<X3<12)= 12 Discrete Random Variable A discrete random variable X has a countable number of possible values. The probability distribution of X x x1 x2 x3 … xk Prob p1 p2 p3 … pk where, 1. Every pi is a between 0 and 1. 2. p1 + p2 +…+ pk = 1. Continuous Random Variable A continuous random variable X takes all values in an interval of numbers. Examples: X11 = Amount of rain in October. X12 = Amount of milk produced by a cow. X13 = Useful life of a bulb. X14 = Height of college students. X15 = Average salary of UWL faculty. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event. Continuous Distributions 1. 2. 3. 4. 5. 6. Normal Distribution Uniform Distribution Chi-squared Distribution T-Distribution F-Distribution Gamma Distribution Homework Exercises: Sec 4.3 : #41, 45, 46, 53, 54, 56. Sec 4.4 : #60, 64, 69, 72, 74, 75, 76. Thank you!