This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Examples: Continuous RV? National daily oil demand % return of an investment portfolio Number of Zipcar rentals today at 34th and Chestnut St. Ratio of US $ to Euro 15 Discrete vs Continuous Random Variables When there are finitely many or countable number of values of a random variable, it is called a discrete random variable . Examples: Discrete RV? Number of spots on a die Yes Number of items purchased by a customer Yes Number of times you pushed snooze on your alarm this morning Yes Outside temperature on February 1st No When a random variable can take any realnumber value (in an open or closed range), it is called a continuous random variable. Examples: Continuous RV? National daily oil demand Yes % return of an investment portfolio Yes Number of Zipcar rentals today at 34th and Chestnut St. No Ratio of US $ to Euro Yes 16 Discrete Random Variables Other Examples: Roll a die twice. Let x be the number of times 4 comes up, then x can be 0, 1, or 2 times. 17 Discrete Random Variables Other Examples: Roll a die twice. Let x be the number of times 4 comes up, then x can be 0, 1, or 2 times. Toss a coin 5 times. Let x be the number of heads, then x = {0, 1, 2, 3, 4, or 5} 18 For any discrete random variable, X Discrete random variable: Probability mass function (PMF) Probability mass function p(x) = Probability the random variable X takes on the value x Properties p(x) ≥ 0 for all x, (p(x) > 0 if x is a possible value of X) p(x) = 1 19 Discrete random variable: Probability mass function (PMF) Example Experiment: Toss 2 coins X = Number of heads in two coin tosses is a random variable Probability distribution of X T T 4 possible outcomes T T H H H H Probability Distribution 0 1 2 x X value Probability 0 1/4 = 0.25 1 2/4 = 0.50 2 1/4 = 0.25 0.50 0.25 Probability 20 Discrete random variable: Cumulative distribution (CDF) Cumulative distribution function (CDF): The probability a random variable X takes on a value less than or equal to x. If X is a nonnegative integer r.v F(x)= Note: P(X>x)= 1 – P(X ≤ x) p(0) + p(1) + … + p(x) F ( x ) = Pr( X ≤ x ) = p ( i ) i ≤ x ∑ 21 1/6 or 0.1667 21/36 = 7/12 = 0.5813 Example: Rolling Two Dice Use “F(x)” for cumulative distribution Rolling Two Dice (Probability Mass Function) 0.000 0.028 0.056 0.084 0.112 0.140 0.168 0.196 2 3 4 5 6 7 8 9 10 11 12 Sum of two dices Probability = = ) 7 ( X P = ≤ = ) 7 ( ) 7 ( X P F Rolling Two Dice (Cumulative Distribution) 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 2 3 4 5 6 7 8 9 10 11 12 Sum of two dices Probability 22...
View
Full Document
 Fall '12
 StephenD.Joyce
 Probability theory, discrete random variable

Click to edit the document details