lect11v7_1up

lect11v7_1up - Stat 104: Quantitative Methods for...

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Stat 104: Quantitative Methods for Economists Class 11: Random Variables, Part I 1
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Turning Letters into Numbers s We’ve been studying statements like P(A) or P(B|A). That is, probability of different events happening. ut many things in the real world are easier s But many things in the real world are easier explained using numerical outcomes. s The concept of random variables enables us to talk about the probability of numerical outcomes. 2
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Random Variables A random variable is a variable that takes on different values with different probabilities, and is denoted by a capital letter. umber of students who walk in late to class ? Example Situations 3 •Number of students who walk in late to class ? The random variable X will take on the values 0,1,2,….66 (66 students in each class) •How many yen will a dollar buy in currency markets a month from today? the random variable Y , representing the number of yen per dollar, might take on any possible value in the interval (100,300).
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Discrete versus Continuous Random Variables A random variable is called discrete if it can assume only a countable number of possible values. A continuous random variable may assume any 4 numerical value within a given interval and thus has an infinite number of possible outcomes. From previous slide: X discrete continuous Y discrete continuous
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A little notation We may get sloppy from time to time, but usually in the rest of this course, we will stress the distinction between a random variable and the values it can take on by following the convention of using capital letters such as X to denote random variables and 5 lowercase letters such as x to denote their values. For example, say we throw a die and let X be the outcome. The random variable X can take the specific values x=1,x=2,. ...,x=6.
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Hey, I thought calculus wasn’t required! Well, its not. It turns out to fully understand continuous random variables, you have to know a little bit about calculus. In particular, you have to know what this symbol means: 6 For this reason, virtually all the results for random variables will be presented from the discrete point of view. Phew. All the same results hold though for continuous random variables.
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A summary before we begin Sample (Data) X Random Variable Histogram Distribution Mean x Expectation E(X) 7 Variance s Variance Var(X) 2
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The probability mass function P X (x) of a discrete random variable expresses the probability that X takes the value x: ) ( ) ( x X P x P X = = The Probability Function 8 Example: X = outcome when we roll a fair die. P P etc X X ( ) / , ( ) / , ... 1 1 6 2 1 6 = = What is P X (17) ??
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We assign probabilities (chances) to all the values that the random variable can take on. There are certain requirements about the
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lect11v7_1up - Stat 104: Quantitative Methods for...

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