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Chapter 8 – Further Applications of Integration

Chapter 8 – Further Applications of Integration. 8.5 Probability. Definitions.

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Chapter 8 – Further Applications of Integration

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  1. Chapter 8 – Further Applications of Integration 8.5 Probability 8.5 Probability

  2. Definitions • Let’s consider the cholesterol level of a person chosen at random from a certain age group or the height of an adult male or female chosen at random. These quantities are called continuous random variable because their values actually range over an interval of real numbers even though they might be recorded only to the nearest integer. 8.5 Probability

  3. Definitions • Every continuous random variable X has a probability density functionf. This means that the probability that X lies between a and b is found by integrating f from a to b. • Because probabilities are measured on a scale from 0 to 1, it follows that 8.5 Probability

  4. Example 1 – pg. 573 #4 • Let if x 0 and f (x) = 0 if x < 0. • Verify that f is a probability density function. • Find P(1 X  2). 8.5 Probability

  5. Average Values • The mean of any probability density function f is defined to be • This mean can be interpreted as the long-run average value of the random variable X. It can also be interpreted as a measure of centrality of the probability density function. 8.5 Probability

  6. Mean • If  is the region that lies under the graph of f, we know from section 8.3 that the x-coordinate of the centroid of  is • So a thin plate in the shape of  balances at a point on the vertical line x = . 8.5 Probability

  7. Median • Another measure of a central probability density function is the median. In general, the median of a probability density function is the number m such that 8.5 Probability

  8. Example 2 – pg. 574 #9 • Suppose the average waiting time for a customer’s call to be answered by a company representative is five minutes. Show that the median waiting time for a phone company is about 3.5 minutes. 8.5 Probability

  9. Normal Distribution • The normal distribution is a continuous probability distribution that often gives a good description of data that cluster around the mean. The probability density function of the random variable X is a member of the family of functions • The positive constant  is the standard deviation. It measures how spread out the values of X are. 8.5 Probability

  10. Normal Distribution • We can see how the graph changes as  changes. • We can say that 8.5 Probability

  11. Example 3 – pg. 574 # 10 • A type of light bulb is labeled as having an average lifetime of 1000 hours. It’s reasonable to model the probability of failure of these bulbs by an exponential density function with  = 1000. Use this model to find the probability that a bulb • fails within the first 200 hours. • burns for more than 800 hours. • What is the median lifetime of these light bulbs? 8.5 Probability

  12. Example 4 – pg. 574 #12 • According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches. • What is the probability that an adult male chosen at random is between 65 inches and 73 inches tall? • What percentage of the adult male population is more than 6 feet tall? 8.5 Probability

  13. Example 5 – pg. 574 #14 • Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation of 12 g. • If the target weight is 500 g, what is the probability that the machine produces a box with less than 480 g of cereal? • Suppose a law states that no more than 5% of a manufacturer’s cereal boxes can contain less than the stated weight of 500 g. At what target weight should the manufacturer set its filling machine? 8.5 Probability

  14. Book Resources • Video Examples • Example 2 – pg. 569 • Example 4 – pg. 571 • Example 5 – pg. 572 • More Videos • Expected values or means • Calculating Probability • Wolfram Demonstrations • Area of a Normal Distribution 8.5 Probability

  15. Web Resources • http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/moreApps/gaussian.html • http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/moreApps/gaussian.html • http://www.youtube.com/watch?v=szjL60gAweE&feature=youtu.be 8.5 Probability

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