Math 145 July 2 2012 Random Variable A

Math 145 July 2, 2012

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. 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 : X 1 = The number of heads. = {3, 2, 2, 2, 1, 1, 1, 0} X 2 = 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: X 3 = Total no. of dots 2 3 4 5 6 7 8 9 10 6 7 8 9 10 11 12

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

More Examples Survey: Random variables : X 6 = Age in years. X 7 = Gender {1=male, 0=female}. X 8 = Height. Medical Studies: Random variables : X 9 = Blood Pressure. X 10 = {1=smoker, 0=non-smoker}.

Probability Distribution Tossing a coin 3 times: Sample Space = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Random variable : X 1 = 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 : X 1 = The number of heads. X Prob. 0 1/8 1 3/8 2 3/8 3 1/8

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)

Rolling a Pair of Dice Random variable: X 3 = Total no. of dots x P 2 2 3 4 5 6 7 8 6 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: X 3 = 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. Pr(X 3<5)= 2. Pr(3<X 3<12)=

Discrete Random Variable A discrete random variable X has a countable number of possible values. The probability distribution of X x Prob x 1 p 1 x 2 p 2 x 3 p 3 where, 1. Every pi is a between 0 and 1. 2. p 1 + p 2 +…+ pk = 1. … … xk pk

Mean of a Discrete R. V. The probability distribution of X x Prob x 1 p 1 x 2 p 2 x 3 p 3 … … 1. Mean ( ) = E(X) = x 1 p 1+x 2 p 2+…+ xkpk 2. Variance ( 2) = V(X) = (x 1 - )2 p 1 + (x 2 )2 p 2 + …+ (xk- ) 2 pk. xk pk

Continuous Random Variable A continuous random variable X takes all values in an interval of numbers. Examples: X 11 = Amount of rain in October. X 12 = Amount of milk produced by a cow. X 13 = Useful life of a light bulb. X 14 = Height of college students. X 15 = 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

Thank you!