MATH 11 Lecture 12: 5117 Lecture notes (12) Handout

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David James Quarfoot

5/1/17 Lecture notes (12) Handout Tuesday, May 2, 2017 1:44 PM Handout, Day 1: Continuous random variables Note: the book doesn’t cover this material well, so a supplement on Continuous Random Variables can be found on Triton Ed. Outcome Probability of the event -$4 1/2 $72 (1/2)(1/6)= 1/12 -$12 (1/2)(5/6)= 5/12 X = number of trials needed to get first success Probability 1 P(1) = p 2 P(2) = qp 3 P(3) = qqp … X P(x) = q p A discrete random variable takes on finitely many values or infinitely many values which are discrete (spaces in between) A continuous random variable is a random quantity that can take on any value on a continuous scale (a smoothinterval of possibilities). Examples: the amount of water you drink in a day, how long you wait for a bus, how far you live from the nearest grocery store. Some awkward questions? - How do I make a probability table for such a situation? - What is the probability of drinking exactly 2.4314gallons of water in a given day? - Is the deominatorof P(A) = equal to infinity? For continuous random variables, we have to change how we present the probability model (no more tables) alter how we seek questions, generalize our definition of probability and rethink what a probability of 0 means From discrete to continuous random variables Suppose we flip a coin 16 times and record how many heads we get. That is let X = Binom (n=16, p=0.5) When we visualize the probability table: An outcomeis more likely if there is more area in the rectangle for that value We also know that sum of the areas of the bars must be 1 We also know that sum of the areas of the bars must be 1 Finally, we see that the bar heights must be at least 0 ( no negative heights) If we momentarilyuse dots, instead of rectangles, the picture suggests how we could generalize just draw curves. When we do, the meaning of the y - axis changes With discrete random variables, two ideas are linked with probability: height on the y-axis and area ( of a rectangle) With continuous random variables, only area is linked to probability Building the new uni
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