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07_Discrete Probability Distributions Part 1

Examples continuous rv national daily oil demand

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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 real-number 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 non-negative 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...
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