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Chapter 8

# Chapter 8 - MSIT 3000 8.1 Expected Value of a Random...

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MSIT 3000 8.1 Expected Value of a Random Variable A variable whose value is based on the outcome of a random event is called a random variable . If we can list all possible outcomes, the random variable is called a discrete random variable . If a random variable can take on any value between two values, it is called a continuous random variable . The mean of a probability distribution is denoted by the parameter, µ, but usually called the Expected Value and denoted E(X) or EV. This expected value reflects not what we’ll observe in a single observation, but rather what we expect for the average in a long run of observations. 103

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Ex: What’s the Expected Number of Home Runs in a Baseball Game? Find the mean of this probability distribution. Think about calculating a mean of this probability distribution. Since P(0) = 0.23, in the long run, you’d expect no home runs in 23% of games. Likewise, P(1) = 0.38, so in the long run, you’d expect one home run in 38% of games. Since the mean equals the total of the observations divided by the sample size, for 100 games, you’d expect a mean of about . . . 104
µ = (0+0+…0) + (1+1+…1) + (2+2+…2) … +5 100 Where you have 23 zero’s , 38 ones, 22 two’s 13 three’s, 4 fours, and 1 five (look back at table). So µ = 1.38 Written slightly differently: µ = 0(0.23) + 1(0.38) + 2(0.22) + 3(0.13) + 4(0.03) + 5(0.01) = 1.38 In general, for a discrete probability distribution, the mean, or Expected Value is: 105 E(x) = EV ( ) x p x μ= = × 0(23) 1(38) 2(22) 3(13) 4(3) 5(1) 139 μ 100 100 100 100 100 100 100 = + + + + + =

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: The probability model for a particular life insurance policy is shown. Find the expected annual payout on a policy. We expect that the insurance company will pay out \$200 per policy per year. (THIS IS AN AVERAGE INTERPRETATION -- WE DO NOT EXPECT TO PAY EACH INDIVIDUAL EXACTLY \$200. ) Ex: The number of dropped calls experienced per day by customers of a particular cell phone provider are 0, 1, 2, or 3. If the probability associated with each of these values is 0.55, 0.30, 0.10 and 0.05, respectively, what is the expected number of dropped calls per day? 106
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Chapter 8 - MSIT 3000 8.1 Expected Value of a Random...

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