Week 7 Wed Oct 13

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

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1 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 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 x p x X E X E X V ) ( ) ( X V X SD 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 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.

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2 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)
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Week 7 Wed Oct 13 - This version was posted after class...

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