Lecture04

Lecture04 - Mean of a random variable The mean of a set of...

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Probability and Sampling Distributions Random variables Section 4.3 (Continued) © 2009 W.H. Freeman and Company Mean of a random variable The mean of a set of observations is their arithmetic average. The mean ! of a random variable X is a weighted average of the possible values of X , reflecting the fact that all outcomes might not be equally likely. Value of X 0 1 2 3 Probability 1/8 3/8 3/8 1/8 HMM HHM MHM HMH MMM MMH MHH HHH A basketball player shoots three free throws. The random variable X is the number of baskets successfully made (“H”). The mean of a random variable X is also called expected value of X . Mean of a discrete random variable For a discrete random variable X with probability distribution ! the mean ! of X is found by multiplying each possible value of X by its probability, and then adding the products. Value of X 0 1 2 3 Probability 1/8 3/8 3/8 1/8 The mean ! of X is ! = (0*1/8) + (1*3/8) + (2*3/8) + (3*1/8) = 12/8 = 3/2 = 1.5 A basketball player shoots three free throws. The random variable X is the number of baskets successfully made. Variance of a random variable The variance and the standard deviation are the measures of spread that accompany the choice of the mean to measure center. The variance " 2 X of a random variable is a weighted average of the squared deviations ( X ! ! X ) 2 of the variable X from its mean ! X . Each outcome is weighted by its probability in order to take into account outcomes that are not equally likely. The larger the variance of X , the more scattered the values of X on average. The positive square root of the variance gives the standard deviation " of X .
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Variance of a discrete random variable For a discrete random variable X with probability distribution ! and mean ! X , the variance " 2 of X is found by multiplying each squared deviation of X by its probability and then adding all the products. Value of X 0 1 2 3 Probability 1/8 3/8 3/8 1/8 The variance " 2 of X is " 2 = 1/8*(0 ! 1.5) 2 + 3/8*(1 ! 1.5) 2 + 3/8*(2 ! 1.5) 2 + 1/8*(3 ! 1.5) 2 = 2*(1/8*9/4) + 2*(3/8*1/4) = 24/32 = 3/4 = .75 A basketball player shoots three free throws. The random variable X is the number of baskets successfully made. ! X = 1.5. Rules for means and variances If X is a random variable and a and b are fixed numbers, then ! a + bX = a + b ! X " 2 a+bX = b 2 " 2 X If X and Y are two independent random variables, then ! X+Y = ! X + ! Y " 2 X+Y = " 2 X + " 2 Y " 2 X-Y = " 2 X + " 2 Y If X and Y are NOT independent but have correlation # , then ! X+Y = ! X + ! Y " 2 X+Y = " 2 X + " 2 Y + 2 #" X " Y " 2 X-Y = " 2 X + " 2 Y - 2 #" X " Y Investment You invest 20% of your funds in Treasury bills and 80% in an “index fund” that represents all U.S. common stocks. Your rate of return over time is proportional to that of the T-bills ( X ) and of the index fund ( Y ), such that R = 0.2 X + 0.8 Y . Based on annual returns between 1950 and 2003:
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This note was uploaded on 12/11/2009 for the course STAT 212 taught by Professor Holt during the Spring '08 term at UVA.

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Lecture04 - Mean of a random variable The mean of a set of...

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