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Some Continuous Probability Distributions

Some Continuous Probability Distributions. By: Prof. Gevelyn B. Itao. Probabilit y and Statistics. Continuous Uniform Distribution. The density function of the continuous uniform random variable X on the interval [A, B] is. Probabilit y and Statistics. Theorem 6.1.

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Some Continuous Probability Distributions

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  1. Some Continuous Probability Distributions By: Prof. Gevelyn B. Itao

  2. Probability and Statistics Continuous Uniform Distribution • The density function of the continuous uniform random variable X on the interval [A, B]is

  3. Probability and Statistics Theorem 6.1 The mean and variance of the continuous uniform distribution f (x; k) are and

  4. Probability and Statistics Continuous Uniform Distribution • Example 5.1: Suppose that a large conference room for a certain company can be reserved for no more than 4 hours. However, the use of the conference room is such that both long and short conferences occur quite often. In fact, it can be assumed that length X of a conference has a uniform distribution on the interval [0, 4]. • What is the probability density function? • What is the probability that any given conference lasts at least 3 hours?

  5. Probability and Statistics Continuous Uniform Distribution Example 5.2: Calculate the mean and variance in prob 1.

  6. Probability and Statistics Normal Distribution Properties of Normal Curve The mode, which is the point on the horizontal axis where the curve is a maximum, occurs at x = 

  7. Probability and Statistics Normal Distribution Properties of Normal Curve The curve is symmetric about a vertical axis through the mean 

  8. Probability and Statistics Normal Distribution Properties of Normal Curve The curve has its points of inflection at x =  , is concave downward if  -  < X <  +  , and is upward otherwise

  9. Probability and Statistics Normal Distribution Properties of Normal Curve The normal curve approaches the horizontal axis asymptotically as we proceed in either direction away from the mean.

  10. Probability and Statistics Normal Distribution Properties of Normal Curve The total area under the curve and above the horizontal axis is equal to 1.

  11. Probability and Statistics Areas Under the Normal Curve

  12. Probability and Statistics Areas Under the Normal Curve

  13. Probability and Statistics Definition 6.1 Standard Normal Distribution • The distribution of a normal random variable with mean zero and variance 1.

  14. Probability and Statistics Areas Under the Normal Curve Example 6.3: Given a standard normal distribution, find the area under the curve that lies to the right of z = 1.84 between z = -1.97 and z = 0.86

  15. Probability and Statistics Areas Under the Normal Curve Example 6.3: Given a standard normal distribution, find the area under the curve that lies to the right of z = 1.84

  16. Probability and Statistics Areas Under the Normal Curve Example 6.3: Given a standard normal distribution, find the area under the curve that lies between z = -1.97 and z = 0.86

  17. Probability and Statistics Areas Under the Normal Curve Example 6.4: Given a standard normal distribution, find the value of k such that P (Z > k) = 0.3015 P (k < Z < -0.18) = 0.4197

  18. Probability and Statistics Areas Under the Normal Curve Example 6.4: Given a standard normal distribution, find the value of k such that P (Z > k) = 0.3015 From table A.3:

  19. Probability and Statistics Areas Under the Normal Curve Example 6.4: Given a standard normal distribution, find the value of k such that P (k < Z < -0.18) = 0.4197 From table A.3:

  20. Probability and Statistics Areas Under the Normal Curve Example 6.5: Given a normal distribution with  = 50 and  = 10, find the probability that X assumes a value between 45 and 62.

  21. Probability and Statistics Areas Under the Normal Curve Example 6.5: Given a normal distribution with  = 50 and  = 10, find the probability that X assumes a value between 45 and 62.

  22. Probability and Statistics Areas Under the Normal Curve Example 6.5: Given a normal distribution with  = 50 and  = 10, find the probability that X assumes a value between 45 and 62.

  23. Probability and Statistics Areas Under the Normal Curve Example 6.6: Given a normal distribution with  = 40 and  = 6, find the value of x that has 45% of the area to the left 14% of the area to the right

  24. Probability and Statistics Areas Under the Normal Curve Example 6.6: Given a normal distribution with  = 40 and  = 6, find the value of x that has 45% of the area to the left From table A.3:

  25. Probability and Statistics Areas Under the Normal Curve Example 6.6: Given a normal distribution with  = 40 and  = 6, find the value of x that has 14% of the area to the right

  26. Probability and Statistics Areas Under the Normal Curve • Example 6.7: A research scientist reports that mice will live an average of 40 months when their diets are sharply restricted and then enriched with vitamins and proteins. • Assuming that the lifetimes of such mice are normally distributed with a standard deviation of 6.3 months, find the probability that a given mouse will live • more than 32 months; • less than 28 months; • between 37 and 49 months.

  27. Probability and Statistics Areas Under the Normal Curve • Example 6.8: The finished inside diameter of a piston ring is normally distributed with a mean of 10 centimeters and a standard deviation of 0.03 centimeter. • What proportion of rings will have inside diameters exceeding 10.075 centimeters? • What is the probability that a piston ring will have an inside diameter between 9.97 and 10.03 centimeters? • Below what value of inside diameter will 15% of the piston rings fall?

  28. Probability and Statistics Areas Under the Normal Curve • Example 6.9: A lawyer commutes daily from his suburban home to his midtown office. The average time for a one-way trip is 24 minutes, with a standard deviation of 3.8 minutes. Assume the distribution of trip times to be normally distributed. • What is the probability that a trip will take at least 1/2 hour? • If the office opens at 9:00 A.M. and he leaves his house at 8:45 A.M. daily, what percentage of the time is he late for work? • If he leaves the house at 8:35 A.M. and coffee is served at the office from 8:50 A.M. until 9:00 A.M., what is the probability that he misses coffee? • Find the length of time above which we find the slowest 15% of the trips. • Find the probability that 2 of the next 3 trips will take at least 1/2 hour.

  29. Probability and Statistics Normal Approximation to the Binomial Theorem 6.2 If X is a binomial random variable with mean  = npand variance 2 = npq, then the limiting form of the distribution of as n  , is the standard normal distribution n (z; 0,1)

  30. Probability and Statistics Normal Approximation to the Binomial • Example 6.10: A process for manufacturing an electronic component is 1% defective. A quality control plan is to select 100 items from the process, and if none are defective, the process continues. Use the normal approximation to the binomial to find • the probability that the process continues for the sampling plan described; • the probability that the process continues even if the process has gone bad (i.e., if the frequency of defective components has shifted to 5.0% defective).

  31. Probability and Statistics Normal Approximation to the Binomial • Example 6.11: A process yields 10% defective items. If 100 items are randomly selected from the process, what is the probability that the number of defectives • exceeds 13? • is less than 8?

  32. Probability and Statistics Gamma Distribution • The continuous random variable X has a gamma distribution, with parameters α and β, if its density function is given by where α > 0 and β > 0.

  33. Probability and Statistics Definition 6.2 The gamma function is defined by where α > 0.

  34. Probability and Statistics Theorem 6.3 The mean and variance of the gamma distribution are µ = αβ σ2 = αβ2 and

  35. Probability and Statistics Normal Approximation to the Binomial • Example 6.12: In a certain city, the daily consumption of electric power, in millions of kilowatt-hours, is a random variable X having a gamma distribution with mean µ= 6 and variance σ2= 12. • Find the values of α and β. • Find the probability that on any given day the daily power consumption will exceed 12 million kilowatthours.

  36. Probability and Statistics Normal Approximation to the Binomial • Example 6.13: Suppose that the time, in hours, taken to repair a heat pump is a random variable X having a gamma distribution with parameters α =2 and β=1/2. What is the probability that the next service call will require • at most 1 hour to repair the heat pump? • at least 2 hours to repair the heat pump?

  37. Probability and Statistics ExponentialDistribution • The continuous random variable X has an exponential distribution, with parameter β, if its density function is given by where β > 0.

  38. Probability and Statistics Theorem 6.4 The mean and variance of the exponential distribution are µ = β σ2 = β2 and

  39. Probability and Statistics Normal Approximation to the Binomial • Example 6.14: The life, in years, of a certain type of electrical switch has an exponential distribution with an average life β = 2. If 100 of these switches are installed in different systems, what is the probability that at most 30 fail during the first year?

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