Math 145 September 24, 2014

Random Variable Two Types: – 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. Two Types: 1. Discrete Random Variable – A discrete random variable has a countable number of possible values (There is a gap between possible values). 2. Continuous Random Variable – A continuous random variable takes all values in an interval of numbers.

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)

Random variable: X3 = Total no. of dots Rolling a Pair of Dice Random variable: X3 = Total no. of dots 2 3 4 5 6 7 8 9 10 11 12

Rolling a Pair of Dice X4 = (positive) difference in the no. of dots 1 2 3 4 5

Rolling a Pair of Dice X5 = Higher of the two. 1 2 3 4 5 6

More Examples Survey: Medical Studies: Random variables : X6 = Age in years. X7 = Gender {1=male, 0=female}. X8 = Height. Medical Studies: 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 1 2 3 Prob. 1/8 3/8

Probability Histogram Tossing a coin 3 times: Random variable : X1 = The number of heads. X 1 2 3 Prob. 1/8 3/8

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)

Random variable: X3 = Total no. of dots Rolling a Pair of Dice Random variable: X3 = Total no. of dots 2 3 4 5 6 7 8 9 10 11 12 x 2 3 4 5 6 7 8 9 10 11 12 P 1/36 2/36 3/36 4/36 5/36 6/36

Random variable: X3 = Total no. of dots Rolling a Pair of Dice Random variable: X3 = Total no. of dots x 2 3 4 5 6 7 8 9 10 11 12 P 1/36 2/36 3/36 4/36 5/36 6/36 1. Pr(X3<5)= 2. Pr(3<X3<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, Every pi is a between 0 and 1. p1 + p2 +…+ pk = 1.

Mean () = E(X) = x1p1+x2p2+…+ xkpk Mean of a Discrete R.V. The probability distribution of X x x1 x2 x3 … xk Prob p1 p2 p3 pk Mean () = E(X) = x1p1+x2p2+…+ xkpk Variance (2) = V(X) = (x1-)2p1 + (x2-)2p2 + …+ (xk-) 2pk .

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 Normal Distribution Uniform Distribution Chi-squared Distribution T-Distribution F-Distribution Gamma Distribution

Thank you!