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Week 7 Wed Oct 13 - This version was posted after class...

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This version was posted after class. WEEK 7 – Wed, Oct 13 Chapter 3. Random Variables and Probability Distributions on the Line . Discrete – Probability mass function . p ( x ) Cumulative distribution function . = = x y y p x X P x F ) ( ) ( ) ( Expectation . ) ( ) ( = = x xp X E μ called the mean of X or simply the expected value of X . For any function h and Y = h ( X ), Y is a random variable with expectation = = ) ( ) ( )) ( ( ) ( x p x h X h E Y E . Variance and Standard Deviation . )] ( [ ) ( ] ) [( ) ( 2 2 2 2 2 2 μ μ μ σ - = - = - = = x p x X E X E X V ) ( ) ( X V X SD = = σ Please note . The units of measure for E ( X ) and SD ( X ) are the same as the units of measure for X . If X is measured in feet , then E ( X ) and SD ( X ) are in units of feet . If X is in units of ($1,000), then E ( X ) and SD ( X ) are in units of ($1,000). The expectation of X , written E ( X ), is a measure of the “center ” of the probability distribution. It is the balance point for the distribution of probability masses positioned at the possible values of X . In the dice example, the plots of the distribution of X = Total on Pair of Fair Dice show by symmetry that the balance point is 7 so that we can conclude that 7 ) ( = = X E μ without computation. For non-symmetric distributions, we can only guess the balance point from the plot. The standard deviation of X, written SD ( X ), is a measure of the “spread ” of the probability distribution. We will build up intuition in regard to this measure as we proceed further in the course. 1
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Chebyshev Inequality . where k is any real number k ≥ 1. Example 1. So, if µ = 50 and σ = 5, Example 2. Suppose that Y is a random variable with µ = E ( Y ) = $20,000 and σ = SD ( Y ) = $5,477. Use Chebyshev to determine a lower bound for the probability P (| Y – $20,000| < $10,000). Ans. Note that the event in question is $10,000 < Y < $30,000. Recall that a z -score tells you how many standard deviations a point is away from the mean; the z -score of $30,000 is ($30,000 - $20,000)/$5,477 = 1.826 and the z -score of $10,000 is ($10,000 - $20,000)/ $5,477 = -1.826. Thus, P($10,000 < Y < $30,000) = P($20,000 – 1.826($5,477) < Y < ($20,000 + 1.826($5,477)) 0.70. SPECIAL PROBABILITY DISTRIBUTIONS A probability distribution is a model for the uncertainty of a random variable. A random variable has as its outcomes a set of real numbers. If the set of outcomes can be listed in a table, the random variable is said to be discrete. Continuous random variables take values across a continuum of real numbers, perhaps, an interval. Certain probability distributions arise so commonly in applications, that they are studied extensively. Examples: BINOMIAL PROBABILITY DISTRIBUTIONS (DISCRETE) GEOMETRIC DISTRIBUTIONS (DISCRETE) HYPERGEOMETRIC PROBABILITY DISTRIBUTIONS (DISCRETE) UNIFORM PROBABILITY DISTRIBUTIONS (CONTINUOUS) EXPONENTIAL PROBABILITY DISTRIBUTIONS (CONTINUOUS) NORMAL PROBABILITY DISTRIBUTIONS (CONTINUOUS) For discrete random variables, the probabilities come in masses. With 2
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p ( x ) denoting P ( X = x ), we call the function p the probability mass function of the random variable X . For continuous random variables, the probability is spread on the real number line through a probability density function, a function f ≥ 0 such that for all a b , ) ( ) ( ) ( ) ( ) ( b X a P b X a P b X a P dx x f b X a P b a = < = < = = < < .
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