Statistics October 6, 2009

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

Rolling a Pair of Dice X 4 = (positive) difference in the no. of dots

Rolling a Pair of Dice X 5 = Higher of the two

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} x0123 Prob.1/83/83/81/8

Probability Histogram Tossing a coin 3 times: Random variable : X 1 = The number of heads. X0123 Pro b. 1/83/83/81/8

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 P1/362/363/364/365/366/365/364/363/362/361/36

Rolling a Pair of Dice Random variable: X 3 = Total no. of dotsx P1/362/363/364/365/366/365/364/363/362/361/36 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 x1x1x1x1 x2x2x2x2 x3x3x3x3… xkxkxkxk Prob p1p1p1p1 p2p2p2p2 p3p3p3p3… pkpkpkpk where, 1.Every p i is a between 0 and 1. 2.p 1 + p 2 +…+ p k = 1.

Mean of a Discrete R.V. The probability distribution of X x x1x1x1x1 x2x2x2x2 x3x3x3x3… xkxkxkxk Prob p1p1p1p1 p2p2p2p2 p3p3p3p3… pkpkpkpk 1.Mean (  ) = E(X) = x 1 p 1 +x 2 p 2 +…+ x k p k 2.Variance (  2 ) = V(X) = (x 1 -  ) 2 p 1 + (x 2 -  ) 2 p 2 + …+ (x k -  ) 2 p k.

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 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.Normal Distribution 2.Uniform Distribution 3.Chi-squared Distribution 4.T-Distribution 5.F-Distribution 6.Gamma Distribution

Thank you!