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Continuous Probability Distributions

Continuous Probability Distributions. f(x). Uniform Probability Distribution Area as a measure of Probability The Normal Curve The Standard Normal Distribution Computing Probabilities for a Standard Normal Distribution. X. Uniform Probability Distribution. NY. Chicago.

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Continuous Probability Distributions

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  1. Continuous Probability Distributions f(x) • Uniform Probability Distribution • Area as a measure of Probability • The Normal Curve • The Standard Normal Distribution • Computing Probabilities for a Standard Normal Distribution X

  2. Uniform Probability Distribution NY Chicago • Consider the random variable x representing the flight time of an airplane traveling from Chicago to NY. • Under normal conditions, flight time is between 120 and 140 minutes. • Because flight time can be any value between 120 and 140 minutes, x is a continuous variable.

  3. Uniform Probability Distribution With every one-minute interval being equally likely, the random variable x is said to have a uniform probability distribution

  4. Uniform Probability Distribution For the flight-time random variable, a = 120 and b = 140

  5. Uniform Probability Density Function for Flight time The shaded area indicates the probability the flight will arrive in the interval between 120 and 140 minutes 120 125 130 135 140

  6. Basic Geometry Remember when we multiply a line segment times a line segment, we get an area

  7. Probability as an Area Question: What is the probability that arrival time will be between 120 and 130 minutes—that is: 10 120 125 130 135 140

  8. Notice that in the continuous case we do not talk of a random variable assuming a specific value. Rather, we talk of the probability that a random variable will assume a value within a given interval.

  9. E(x) and Var(x) for the Uniform Continuous Distribution Applying these formulas to the example of flight times of Chicago to NY, we have: Thus

  10. Normal Probability Distribution The normal distribution is by far the most important distribution for continuous random variables. It is widely used for making statistical inferences in both the natural and social sciences.

  11. Normal Probability Distribution • It has been used in a wide variety of applications: Heights of people Scientific measurements

  12. Normal Probability Distribution • It has been used in a wide variety of applications: Test scores • Amounts • of rainfall

  13. The Normal Distribution Where: μ is the mean σis the standard deviation  = 3.1459 e = 2.71828

  14. Normal Probability Distribution • Characteristics The distribution is symmetric, and is bell-shaped. x

  15. Normal Probability Distribution • Characteristics The entire family of normal probability distributions is defined by itsmeanm and its standard deviations . Standard Deviation s x Meanm

  16. Normal Probability Distribution • Characteristics The highest point on the normal curve is at the mean, which is also the median and mode. x

  17. Normal Probability Distribution • Characteristics The mean can be any numerical value: negative, zero, or positive. x -10 0 20

  18. Normal Probability Distribution • Characteristics The standard deviation determines the width of the curve: larger values result in wider, flatter curves. s = 15 s = 25 x

  19. Normal Probability Distribution • Characteristics Probabilities for the normal random variable are given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and .5 to the right). .5 .5 x

  20. The Standard Normal Distribution The Standard Normal Distribution is a normal distribution with the special properties that is mean is zero and its standard deviation is one.

  21. Standard Normal Probability Distribution The letter z is used to designate the standard normal random variable. s = 1 z 0

  22. Cumulative Probability Probability that z≤ 1 is the area under the curve to the left of 1. z 1 0

  23. What is P(z ≤ 1)? To find out, use the Cumulative Probabilities Table for the Standard Normal Distribution

  24. Exercise 1 • Answer: • .9931 • 1-.9931=.0069 • What is P(z≤2.46)? • What is P(z≥2.46)? z 2.46

  25. Exercise 2 • Answer: • 1-.9015=.0985 • .9015 • What is P(z≤-1.29)? • What is P(z≥-1.29)? Red-shadedarea isequal togreen- shadedarea Note that: -1.29 z 1.29 Note that, because of the symmetry, the area to the left of -1.29 is the same as the area to the right of 1.29

  26. Exercise 3 What is P(.00≤ z≤1.00)? P(.00≤ z≤1.00)=.3413 0 1 z

  27. Exercise 4 What is P(-1.67 ≥ z ≥1.00)? P(-1.67≤ z≤1.00)=.7938 Thus P(-1.67 ≥ z ≥1.00) =1 - .7938 = .2062 -1.67 0 1 z

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