06-Random Variables complete


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Unformatted text preview: ach class. The curve also provides a visual image of proportion through its area. If we could get the equation of this smoothed curve, we would have a simple and somewhat accurate summary of the distribution of the response. The picture at the right shows a smoothed curve that is symmetric and bell shaped, even though the underlying histogram is only approximately symmetric. If the data came from a representative sample, the smooth curve could serve as a model, that is, as the probability distribution for the continuous response for the population. So the probability distribution of a continuous random variable is described by a density curve. The probability of an event is the area under the curve for the values of X that make up the event. The probability model for a continuous random variable assigns probabilities to intervals. Definition: A curve (or function) is called a Probability Density Curve if: 1. It lies on or above the horizontal axis. 2. Total area under the curve is equal to 1. KEY IDEA: AREA under a density curve over a range of values corresponds to the PROBABILITY that the random variable X takes on a value in that range. 51 Try It! Some Probability Density Curves I. A density curve for modeling income for employed adults (in $1000s) for a city. Income ($1000s) How would you use the above density curve to estimate the probability of a randomly selected employed adult from this city having an income between $30,000 and $40,000? Find the area under this density between 30 and 40. II. Consider the following curve: 1/20 30 50 a. Is this a density curve? Why? Check the properties – yes! (area=bxh = 20(1/20) = 1) b. If yes, find the probability of observing a response that is less than 35. Area = bxh = 5(1/20) = 5/20 = ¼ or 0.25. Note: P(X < 35) is the same as P(X 35) c. What does the value of 35 correspond to for this distribution? the value of 35 = Q1 52 Try it! Checkout time at a store Let X be the checkout time at a store, which is a random variable that is uniformly distributed on values between 5 and 20 minutes. That is, X is U(5, 20). a. What does the density look like? Sketch it and include a value on the y‐axis. Density density 1/15 - 0 0 55 10 10 15 15 20 20 X=time to check out (minutes) b. What is the probability a person will take more than 10 minutes to check out? P(X > 10) = 10(1/15) = 2/3 = 0.67 c. Given a person has already spent 10 minutes checking out, what is the probability they will take no more than 5 additional minutes to check out? From picture can see it will be ½. Optional way to find P(X 15 | X > 10) = P(X 15 and X > 10) = 5/15 = 5/10 = ½. P(X > 10) 10/15 d. What is the expected time to check out at this store? E(X) = = 12.5 minutes (balancing pt = midpt for symmetric distribution) Definition: Mean of a continuous random variable. Expected Value or Mean = Balancing point of the density curve E(X) = (Sometimes one would need calculus/integration to find it ‐‐ integral instead of sums) There are many...
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