The mean of any discrete random variable is an

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The mean of any discrete random variable is an average of the possible outcomes, with each outcome weighted by its probability. Mean of a Discrete Random Variable Suppose that X is a discrete random variable whose probability distribution is Value: x 1 x 2 x 3 Probability: p 1 p 2 p 3 To find the mean (expected value) of X , multiply each possible value by its probability, then add all the products: 𝜇 𝑥 = 𝐸 𝑋 = 𝑥 1 𝑝 1 + 𝑥 2 𝑝 2 + 𝑥 3 𝑝 3 + ⋯ = 𝑥 𝑖 𝑝 𝑖
Example: Babies Health at Birth 22 The probability distribution for X = Apgar scores is shown below: a. Show that the probability distribution for X is legitimate. b. Make a histogram of the probability distribution. Describe what you see. c. Apgar scores of 7 or higher indicate a healthy baby. What is P ( X 7)? Value: 0 1 2 3 4 5 6 7 8 9 10 Probability: 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053 (a) All probabilities are between 0 and 1, and they add up to 1. This is a legitimate probability distribution. (b) The left-skewed shape of the distribution suggests a randomly selected newborn will have an Apgar score at the high end of the scale. There is a small chance of getting a baby with a score of 5 or lower. (c) P ( X ≥ 7) = .908. We d have a 91% chance of randomly choosing a healthy baby.
Example: Apgar Scores―What’ s Typical? 23 Consider the random variable X = Apgar Score. Compute the mean of the random variable X and interpret it in context. Value: 0 1 2 3 4 5 6 7 8 9 10 Probability: 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053 𝜇 𝑥 = 𝐸 𝑋 = 𝑥 𝑖 𝑝 𝑖 = 0 0.001 + 1 0.006 + 2 0.007 + ⋯ + (10)(0.053) = 8.128 The mean Apgar score of a randomly selected newborn is 8.128. This is the long-term average Apgar score of many, many randomly chosen babies. Note: The expected value does not need to be a possible value of X or an integer! It is a long-term average over many repetitions.
Statistical Estimation 24 Suppose we would like to estimate an unknown μ . We could select an SRS and base our estimate on the sample mean. However, a different SRS would probably yield a different sample mean. This basic fact is called sampling variability : The value of a statistic varies in repeated random sampling. To make sense of sampling variability, we ask, What would happen if we took many samples? Population Sample Sample Sample Sample Sample Sample Sample Sample
The Law of Large Numbers 25 . of values different produce would samples random different all, After ? of estimate accurate an be can How x μ x . m parameter the to closer and closer get to guaranteed is statistic the samples, larger and larger taking on keep we If x Draw independent observations at random from any population with finite mean μ . The law of large numbers says that, as the number of observations drawn increases, the sample mean of the observed values gets closer and closer to the mean μ of the population. Many people incorrectly believe the law of small numbers , which says that we expect even short sequences of random events to show the kind of average behavior that, in fact, appears only in the long run.

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