194 CHAPTER 4 Discrete Probability Distributions Mean, Variance, and Standard Deviation You can measure the center of a probability distribution with its mean and measure the variability with its variance and standard deviation. The mean of a discrete random variable is defined as follows. The mean of a discrete random variable is given by m = ΣxP1x2. Each value of x is multiplied by its corresponding probability and the products are added. Mean of a Discrete Random Variable The mean of a random variable represents the “theoretical average” of a probability experiment and sometimes is not a possible outcome. If the experiment were performed many thousands of times, then the mean of all the outcomes would be close to the mean of the random variable. Finding the Mean of a Probability Distribution The probability distribution for the personality inventory test for passive-aggressive traits discussed in Example 2 is shown below. Find the mean score. Score, x 1 2 3 4 5 Probability, P1x2 0.16 0.22 0.28 0.20 0.14 SOLUTION Use a table to organize your work, as shown below. Mean x P1x2 xP1x2 1 0.16 110.162 = 0.16 2 0.22 210.222 = 0.44 3 0.28 310.282 = 0.84 4 0.20 410.202 = 0.80 5 0.14 510.142 = 0.70 ΣP1x2 = 1 ΣxP1x2 = 2.94 ≈ 2.9 From the table, you can see that the mean score is m = 2.94 ≈ 2.9. Note that the mean is rounded to one more decimal place than the possible values of the random variable x. Interpretation Recall that a score of 3 represents an individual who exhibits neither passive nor aggressive traits and the mean is slightly less than 3. So, the mean personality trait is neither extremely passive nor extremely aggressive, but is slightly closer to passive. TRY IT YOURSELF 5 Find the mean of the probability distribution you constructed in Try It Yourself 2. What can you conclude? Answer: Page A38 EXAMPLE 5 Study Tip The mean in Example 5 was rounded to one decimal place because the mean of a probability distribution should be rounded to one more decimal place than was used for the random variable x. This round-off rule is also used for the variance and standard deviation of a probability distribution.
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