Ch6-Notes-Sol.pdf - STAT 3021 Spring 2018 Chapter 6 Some...

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STAT 3021 Spring 2018 Chapter 6. Some Continuous Probability Distributions Recall: If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a continuous sample space. A random variable is called a continuous random variable if it can take on values on a continuous scale. Examples of continuous random variable are: The amount of rain, in inches, that falls in a randomly selected storm. The height of a randomly selected person. The length of time to play 18 holes of golf. Recall: Definition 3.3 6.1 Continuous Uniform Distribution Example 1. (Exercise 3.17 on page 92) A continuous variable X that can assume values between x = 1 and x = b has a density function given by f ( x ) = 1 / 2. (a) Find b . R b 1 1 2 dx = 1 2 x b 1 = 1 2 ( b - 1) = 1 ) b = 3 (b) Find P ( X < 1 . 5). P ( X < 1 . 5) = R 1 . 5 1 1 2 dx = x 2 1 . 5 1 = 0 . 25 (c) Find P ( X 2 | X > 1 . 5). P ( X 2 | X > 1 . 5) = P (1 . 5 <X 2) P ( X> 1 . 5) = P (1 . 5 <X 2) 1 - P ( X 1 . 5) = R 2 1 . 5 1 2 dx 1 - 0 . 25 = 0 . 25 0 . 75 = x 2 | 2 1 . 5 1 - 0 . 25 = 0 . 25 0 . 75 = 1 3 1
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STAT 3021 Spring 2018 (d) Find x such that P ( X x ) = 0 . 75. P ( X x ) = R x 1 1 2 dt = t 2 x 1 = x - 1 2 = 0 . 75 ) x = 2 . 5 Definition: Uniform Distribution Theorem 6.1 Mean and Variance of the uniform distribution Example 2. Let X unif(-3, 5). Find (a) E ( X ) E ( X ) = - 3+5 2 = 1 (b) E [( X - 1) 2 ] E [( X - 1) 2 ] = V ar ( X ) = (5+3) 2 12 = 64 12 = 16 3 2
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STAT 3021 Spring 2018 6.2 Normal Distribution The distributions of many quantitative random variables are well-approximated by the normal distribution. Normal distributions are specified by μ and σ . Once μ and σ are specified, the normal curve is complete determined. μ : measures the center of the normal distribution.
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