07_Discrete Probability Distributions Part 1

# Examples continuous rv national daily oil demand

• Notes
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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

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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.

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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

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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

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21 1/6 or 0.1667 21/36 = 7/12 = 0.5813 Example: Rolling Two Dice Use “F(x)” for cumulative distribution
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• Fall '12
• StephenD.Joyce
• Probability theory, discrete random variable

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