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Chris Morgan, MATH G160 csmorgan@purdue February 24, 2012 Lecture 18

Chris Morgan, MATH G160 csmorgan@purdue.edu February 24, 2012 Lecture 18. Chapter 6.1: Unifrom Distributions. Uniform Distribution. • When we worked with discrete RV’s, we came across many variables for which each outcome was equally likely such as rolling a die or flipping a coin.

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Chris Morgan, MATH G160 csmorgan@purdue February 24, 2012 Lecture 18

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  1. Chris Morgan, MATH G160 csmorgan@purdue.edu February 24, 2012 Lecture 18 Chapter 6.1: Unifrom Distributions

  2. Uniform Distribution • When we worked with discrete RV’s, we came across many variables for which each outcome was equally likelysuch as rolling a die or flipping a coin. • In the continuous case, this type of situation is called a uniformrandom variable. • A uniform random variable is a continuous random variable for which every outcome in an interval is equally likely. NOTE: The probability that X falls in a subinterval only depends on the length (not position) of the subinterval.

  3. Uniform Distribution Examples: - X is a random variable on the closed interval from [0,1] - X is the exact time that someone will sneeze during class - often used when writing random number generators in computer science - any time we have no idea when something will occur, it is equally likely to happen at any time, so we will use uniform (similar to how we used p=0.5 previously)

  4. Uniform Distribution Notation: X ~ Uni(a, b) where: a is the lower bound, and b is the upper bound PDF: f(x) = for a ≤ x ≤ b CDF: F(x) = for a ≤ x ≤ b

  5. Uniform PDF

  6. Uniform CDF

  7. Uniform Distribution Expected Value: Variance:

  8. Uniform Example #1 Suppose you are going out for the evening with friends and they ask you to be ready to leave by 9:00pm. Your friends will arrive at a time T uniformly distributed between 9:00pm and 9:30pm. State the distribution and Parameters of T:

  9. Uniform Example #1a What is the probability that you’ll have to wait for more than 20 minutes for your friends? Remember: A CDF only tells us the probability to the left of X, so when we are looking for a value greater than X, we need to find the compliment (subtract from 1)

  10. Uniform Example #1b If at 9:20pm your friends have no yet arrived, what is the probability that you have to wait at least 5 more minutes?

  11. Uniform Example #1c What is the probability that your friends will arrive at exactly 9:25pm? Why?? - REMEMBER: with a a continuous distribution:

  12. Uniform Example #1d Find the mean and variance of wait time.

  13. Uniform Example #2a Assume that the time it takes a students to walk from one class to the next class on Purdue’s campus ranges uniformly from 0 to 15 minutes. What is the probability that a student will be late to his or her next class, if there is a 10-minute break between classes?

  14. Uniform Example #2b What is the probability that a student will make it to class in exactly five minutes?

  15. Uniform Example #2c What is the probability that it will take a student between 5 and 13 minutes to get to class, given it will take them more than four minutes to get to class?

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